9 - Julia is fast.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Julia is fast\n",
    "\n",
    "Very often, benchmarks are used to compare languages.  These benchmarks can lead to long discussions, first as to exactly what is being benchmarked and secondly what explains the differences.  These simple questions can sometimes get more complicated than you at first might imagine.\n",
    "\n",
    "The purpose of this notebook is for you to see a simple benchmark for yourself.  One can read the notebook and see what happened on the author's Macbook Pro with a 4-core Intel Core I7, or run the notebook yourself.\n",
    "\n",
    "(This material began life as a wonderful lecture by Steven Johnson at MIT: https://github.com/stevengj/18S096/blob/master/lectures/lecture1/Boxes-and-registers.ipynb.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Outline of this notebook\n",
    "\n",
    "- Define the sum function\n",
    "- Implementations & benchmarking of sum in...\n",
    "    - C (hand-written)\n",
    "    - C (hand-written with -ffast-math)\n",
    "    - python (built-in)\n",
    "    - python (numpy)\n",
    "    - python (hand-written)\n",
    "    - Julia (built-in)\n",
    "    - Julia (hand-written)\n",
    "    - Julia (hand-written with SIMD)\n",
    "- Summary of benchmarks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# `sum`: An easy enough function to understand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the  **sum** function `sum(a)`, which computes\n",
    "$$\n",
    "\\mathrm{sum}(a) = \\sum_{i=1}^n a_i,\n",
    "$$\n",
    "where $n$ is the length of `a`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000000-element Vector{Float64}:\n",
       " 0.606379640107222\n",
       " 0.484698866550308\n",
       " 0.3922759781791081\n",
       " 0.7450457529090424\n",
       " 0.5144845015404576\n",
       " 0.29332973122792627\n",
       " 0.4587491906345028\n",
       " 0.1482800720306915\n",
       " 0.6465201369279091\n",
       " 0.06138124692053304\n",
       " 0.6445079369364795\n",
       " 0.5400071393152944\n",
       " 0.7567904000085415\n",
       " ⋮\n",
       " 0.9476614236670423\n",
       " 0.2510283977806267\n",
       " 0.4943419133627476\n",
       " 0.49283400697370205\n",
       " 0.8764599937221115\n",
       " 0.14497273281265077\n",
       " 0.05993865726811931\n",
       " 0.7233953653348673\n",
       " 0.807112804894101\n",
       " 0.48021160468689394\n",
       " 0.6019130540008734\n",
       " 0.3849815924157446"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = rand(10^7) # 1D vector of random numbers, uniform on [0,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035264868e6"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expected result is 0.5 * 10^7, since the mean of each entry is 0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Benchmarking a few ways in a few languages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.004998 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "4.999754035264868e6"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.004459 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "4.999754035264868e6"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.004552 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "4.999754035264868e6"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `@time` macro can yield noisy results, so it's not our best choice for benchmarking!\n",
    "\n",
    "Luckily, Julia has a `BenchmarkTools.jl` package to make benchmarking easy and accurate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.add(\"BenchmarkTools\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "using BenchmarkTools  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  1. The C language\n",
    "\n",
    "C is often considered the gold standard: difficult on the human, nice for the machine. Getting within a factor of 2 of C is often satisfying. Nonetheless, even within C, there are many kinds of optimizations possible that a naive C writer may or may not get the advantage of.\n",
    "\n",
    "The current author does not speak C, so he does not read the cell below, but is happy to know that you can put C code in a Julia session, compile it, and run it. Note that the `\"\"\"` wrap a multi-line string."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: redefinition of constant Clib. This may fail, cause incorrect answers, or produce other errors.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "c_sum (generic function with 1 method)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Libdl\n",
    "C_code = \"\"\"\n",
    "#include <stddef.h>\n",
    "double c_sum(size_t n, double *X) {\n",
    "    double s = 0.0;\n",
    "    for (size_t i = 0; i < n; ++i) {\n",
    "        s += X[i];\n",
    "    }\n",
    "    return s;\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "const Clib = tempname()   # make a temporary file\n",
    "\n",
    "\n",
    "# compile to a shared library by piping C_code to gcc\n",
    "# (works only if you have gcc installed):\n",
    "\n",
    "open(`gcc -fPIC -O3 -msse3 -xc -shared -o $(Clib * \".\" * Libdl.dlext) -`, \"w\") do f\n",
    "    print(f, C_code) \n",
    "end\n",
    "\n",
    "# define a Julia function that calls the C function:\n",
    "c_sum(X::Array{Float64}) = ccall((\"c_sum\", Clib), Float64, (Csize_t, Ptr{Float64}), length(X), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035265293e6"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a) ≈ sum(a) # type \\approx and then <TAB> to get the ≈ symbolb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.246830940246582e-7"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a) - sum(a)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "isapprox (generic function with 9 methods)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "≈  # alias for the `isapprox` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[0m\u001b[1mi\u001b[22m\u001b[0m\u001b[1ms\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\u001b[0m\u001b[1mp\u001b[22m\u001b[0m\u001b[1mp\u001b[22m\u001b[0m\u001b[1mr\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1mx\u001b[22m\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\n",
       "\\end{verbatim}\n",
       "Inexact equality comparison. Two numbers compare equal if their relative distance \\emph{or} their absolute distance is within tolerance bounds: \\texttt{isapprox} returns \\texttt{true} if \\texttt{norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))}. The default \\texttt{atol} is zero and the default \\texttt{rtol} depends on the types of \\texttt{x} and \\texttt{y}. The keyword argument \\texttt{nans} determines whether or not NaN values are considered equal (defaults to false).\n",
       "\n",
       "For real or complex floating-point values, if an \\texttt{atol > 0} is not specified, \\texttt{rtol} defaults to the square root of \\href{@ref}{\\texttt{eps}} of the type of \\texttt{x} or \\texttt{y}, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significant digits. Otherwise, e.g. for integer arguments or if an \\texttt{atol > 0} is supplied, \\texttt{rtol} defaults to zero.\n",
       "\n",
       "The \\texttt{norm} keyword defaults to \\texttt{abs} for numeric \\texttt{(x,y)} and to \\texttt{LinearAlgebra.norm} for arrays (where an alternative \\texttt{norm} choice is sometimes useful). When \\texttt{x} and \\texttt{y} are arrays, if \\texttt{norm(x-y)} is not finite (i.e. \\texttt{±Inf} or \\texttt{NaN}), the comparison falls back to checking whether all elements of \\texttt{x} and \\texttt{y} are approximately equal component-wise.\n",
       "\n",
       "The binary operator \\texttt{≈} is equivalent to \\texttt{isapprox} with the default arguments, and \\texttt{x ≉ y} is equivalent to \\texttt{!isapprox(x,y)}.\n",
       "\n",
       "Note that \\texttt{x ≈ 0} (i.e., comparing to zero with the default tolerances) is equivalent to \\texttt{x == 0} since the default \\texttt{atol} is \\texttt{0}.  In such cases, you should either supply an appropriate \\texttt{atol} (or use \\texttt{norm(x) ≤ atol}) or rearrange your code (e.g. use \\texttt{x ≈ y} rather than \\texttt{x - y ≈ 0}).   It is not possible to pick a nonzero \\texttt{atol} automatically because it depends on the overall scaling (the \"units\") of your problem: for example, in \\texttt{x - y ≈ 0}, \\texttt{atol=1e-9} is an absurdly small tolerance if \\texttt{x} is the \\href{https://en.wikipedia.org/wiki/Earth_radius}{radius of the Earth} in meters, but an absurdly large tolerance if \\texttt{x} is the \\href{https://en.wikipedia.org/wiki/Bohr_radius}{radius of a Hydrogen atom} in meters.\n",
       "\n",
       "\\begin{quote}\n",
       "\\textbf{compat}\n",
       "\n",
       "Julia 1.6\n",
       "\n",
       "Passing the \\texttt{norm} keyword argument when comparing numeric (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\\end{quote}\n",
       "\\section{Examples}\n",
       "\\begin{verbatim}\n",
       "julia> isapprox(0.1, 0.15; atol=0.05)\n",
       "true\n",
       "\n",
       "julia> isapprox(0.1, 0.15; rtol=0.34)\n",
       "true\n",
       "\n",
       "julia> isapprox(0.1, 0.15; rtol=0.33)\n",
       "false\n",
       "\n",
       "julia> 0.1 + 1e-10 ≈ 0.1\n",
       "true\n",
       "\n",
       "julia> 1e-10 ≈ 0\n",
       "false\n",
       "\n",
       "julia> isapprox(1e-10, 0, atol=1e-8)\n",
       "true\n",
       "\n",
       "julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`\n",
       "true\n",
       "\\end{verbatim}\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "isapprox(x; kwargs...) / ≈(x; kwargs...)\n",
       "\\end{verbatim}\n",
       "Create a function that compares its argument to \\texttt{x} using \\texttt{≈}, i.e. a function equivalent to \\texttt{y -> y ≈ x}.\n",
       "\n",
       "The keyword arguments supported here are the same as those in the 2-argument \\texttt{isapprox}.\n",
       "\n",
       "\\begin{quote}\n",
       "\\textbf{compat}\n",
       "\n",
       "Julia 1.5\n",
       "\n",
       "This method requires Julia 1.5 or later.\n",
       "\n",
       "\\end{quote}\n"
      ],
      "text/markdown": [
       "```\n",
       "isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\n",
       "```\n",
       "\n",
       "Inexact equality comparison. Two numbers compare equal if their relative distance *or* their absolute distance is within tolerance bounds: `isapprox` returns `true` if `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The default `atol` is zero and the default `rtol` depends on the types of `x` and `y`. The keyword argument `nans` determines whether or not NaN values are considered equal (defaults to false).\n",
       "\n",
       "For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to the square root of [`eps`](@ref) of the type of `x` or `y`, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significant digits. Otherwise, e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.\n",
       "\n",
       "The `norm` keyword defaults to `abs` for numeric `(x,y)` and to `LinearAlgebra.norm` for arrays (where an alternative `norm` choice is sometimes useful). When `x` and `y` are arrays, if `norm(x-y)` is not finite (i.e. `±Inf` or `NaN`), the comparison falls back to checking whether all elements of `x` and `y` are approximately equal component-wise.\n",
       "\n",
       "The binary operator `≈` is equivalent to `isapprox` with the default arguments, and `x ≉ y` is equivalent to `!isapprox(x,y)`.\n",
       "\n",
       "Note that `x ≈ 0` (i.e., comparing to zero with the default tolerances) is equivalent to `x == 0` since the default `atol` is `0`.  In such cases, you should either supply an appropriate `atol` (or use `norm(x) ≤ atol`) or rearrange your code (e.g. use `x ≈ y` rather than `x - y ≈ 0`).   It is not possible to pick a nonzero `atol` automatically because it depends on the overall scaling (the \"units\") of your problem: for example, in `x - y ≈ 0`, `atol=1e-9` is an absurdly small tolerance if `x` is the [radius of the Earth](https://en.wikipedia.org/wiki/Earth_radius) in meters, but an absurdly large tolerance if `x` is the [radius of a Hydrogen atom](https://en.wikipedia.org/wiki/Bohr_radius) in meters.\n",
       "\n",
       "!!! compat \"Julia 1.6\"\n",
       "    Passing the `norm` keyword argument when comparing numeric (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\n",
       "# Examples\n",
       "\n",
       "```jldoctest\n",
       "julia> isapprox(0.1, 0.15; atol=0.05)\n",
       "true\n",
       "\n",
       "julia> isapprox(0.1, 0.15; rtol=0.34)\n",
       "true\n",
       "\n",
       "julia> isapprox(0.1, 0.15; rtol=0.33)\n",
       "false\n",
       "\n",
       "julia> 0.1 + 1e-10 ≈ 0.1\n",
       "true\n",
       "\n",
       "julia> 1e-10 ≈ 0\n",
       "false\n",
       "\n",
       "julia> isapprox(1e-10, 0, atol=1e-8)\n",
       "true\n",
       "\n",
       "julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`\n",
       "true\n",
       "```\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "isapprox(x; kwargs...) / ≈(x; kwargs...)\n",
       "```\n",
       "\n",
       "Create a function that compares its argument to `x` using `≈`, i.e. a function equivalent to `y -> y ≈ x`.\n",
       "\n",
       "The keyword arguments supported here are the same as those in the 2-argument `isapprox`.\n",
       "\n",
       "!!! compat \"Julia 1.5\"\n",
       "    This method requires Julia 1.5 or later.\n",
       "\n"
      ],
      "text/plain": [
       "\u001b[36m  isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\u001b[39m\n",
       "\n",
       "  Inexact equality comparison. Two numbers compare equal if their relative\n",
       "  distance \u001b[4mor\u001b[24m their absolute distance is within tolerance bounds: \u001b[36misapprox\u001b[39m\n",
       "  returns \u001b[36mtrue\u001b[39m if \u001b[36mnorm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))\u001b[39m. The\n",
       "  default \u001b[36matol\u001b[39m is zero and the default \u001b[36mrtol\u001b[39m depends on the types of \u001b[36mx\u001b[39m and \u001b[36my\u001b[39m.\n",
       "  The keyword argument \u001b[36mnans\u001b[39m determines whether or not NaN values are\n",
       "  considered equal (defaults to false).\n",
       "\n",
       "  For real or complex floating-point values, if an \u001b[36matol > 0\u001b[39m is not specified,\n",
       "  \u001b[36mrtol\u001b[39m defaults to the square root of \u001b[36meps\u001b[39m of the type of \u001b[36mx\u001b[39m or \u001b[36my\u001b[39m, whichever is\n",
       "  bigger (least precise). This corresponds to requiring equality of about half\n",
       "  of the significant digits. Otherwise, e.g. for integer arguments or if an\n",
       "  \u001b[36matol > 0\u001b[39m is supplied, \u001b[36mrtol\u001b[39m defaults to zero.\n",
       "\n",
       "  The \u001b[36mnorm\u001b[39m keyword defaults to \u001b[36mabs\u001b[39m for numeric \u001b[36m(x,y)\u001b[39m and to \u001b[36mLinearAlgebra.norm\u001b[39m\n",
       "  for arrays (where an alternative \u001b[36mnorm\u001b[39m choice is sometimes useful). When \u001b[36mx\u001b[39m\n",
       "  and \u001b[36my\u001b[39m are arrays, if \u001b[36mnorm(x-y)\u001b[39m is not finite (i.e. \u001b[36m±Inf\u001b[39m or \u001b[36mNaN\u001b[39m), the\n",
       "  comparison falls back to checking whether all elements of \u001b[36mx\u001b[39m and \u001b[36my\u001b[39m are\n",
       "  approximately equal component-wise.\n",
       "\n",
       "  The binary operator \u001b[36m≈\u001b[39m is equivalent to \u001b[36misapprox\u001b[39m with the default arguments,\n",
       "  and \u001b[36mx ≉ y\u001b[39m is equivalent to \u001b[36m!isapprox(x,y)\u001b[39m.\n",
       "\n",
       "  Note that \u001b[36mx ≈ 0\u001b[39m (i.e., comparing to zero with the default tolerances) is\n",
       "  equivalent to \u001b[36mx == 0\u001b[39m since the default \u001b[36matol\u001b[39m is \u001b[36m0\u001b[39m. In such cases, you should\n",
       "  either supply an appropriate \u001b[36matol\u001b[39m (or use \u001b[36mnorm(x) ≤ atol\u001b[39m) or rearrange your\n",
       "  code (e.g. use \u001b[36mx ≈ y\u001b[39m rather than \u001b[36mx - y ≈ 0\u001b[39m). It is not possible to pick a\n",
       "  nonzero \u001b[36matol\u001b[39m automatically because it depends on the overall scaling (the\n",
       "  \"units\") of your problem: for example, in \u001b[36mx - y ≈ 0\u001b[39m, \u001b[36matol=1e-9\u001b[39m is an\n",
       "  absurdly small tolerance if \u001b[36mx\u001b[39m is the radius of the Earth\n",
       "  (https://en.wikipedia.org/wiki/Earth_radius) in meters, but an absurdly\n",
       "  large tolerance if \u001b[36mx\u001b[39m is the radius of a Hydrogen atom\n",
       "  (https://en.wikipedia.org/wiki/Bohr_radius) in meters.\n",
       "\n",
       "\u001b[39m\u001b[1m  │ \u001b[22m\u001b[39m\u001b[1mJulia 1.6\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m  Passing the \u001b[36mnorm\u001b[39m keyword argument when comparing numeric\n",
       "\u001b[39m\u001b[1m  │\u001b[22m  (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\u001b[1m  Examples\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> isapprox(0.1, 0.15; atol=0.05)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox(0.1, 0.15; rtol=0.34)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox(0.1, 0.15; rtol=0.33)\u001b[39m\n",
       "\u001b[36m  false\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> 0.1 + 1e-10 ≈ 0.1\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> 1e-10 ≈ 0\u001b[39m\n",
       "\u001b[36m  false\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox(1e-10, 0, atol=1e-8)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  isapprox(x; kwargs...) / ≈(x; kwargs...)\u001b[39m\n",
       "\n",
       "  Create a function that compares its argument to \u001b[36mx\u001b[39m using \u001b[36m≈\u001b[39m, i.e. a function\n",
       "  equivalent to \u001b[36my -> y ≈ x\u001b[39m.\n",
       "\n",
       "  The keyword arguments supported here are the same as those in the 2-argument\n",
       "  \u001b[36misapprox\u001b[39m.\n",
       "\n",
       "\u001b[39m\u001b[1m  │ \u001b[22m\u001b[39m\u001b[1mJulia 1.5\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m  This method requires Julia 1.5 or later."
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?isapprox"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now benchmark the C code directly from Julia:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 406 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m11.762 ms\u001b[22m\u001b[39m … \u001b[35m 14.249 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m12.274 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m12.314 ms\u001b[22m\u001b[39m ± \u001b[32m385.221 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▂\u001b[39m▂\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m▄\u001b[34m▂\u001b[39m\u001b[32m▄\u001b[39m\u001b[39m█\u001b[39m▂\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▆\u001b[39m▆\u001b[39m▆\u001b[39m▅\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m▇\u001b[39m▆\u001b[39m▆\u001b[39m█\u001b[39m█\u001b[39m█\u001b[34m█\u001b[39m\u001b[32m█\u001b[39m\u001b[39m█\u001b[39m█\u001b[39m▆\u001b[39m▄\u001b[39m▅\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▅\u001b[39m▄\u001b[39m▆\u001b[39m▅\u001b[39m▄\u001b[39m▃\u001b[39m▆\u001b[39m▅\u001b[39m▃\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m▃\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m \u001b[39m▄\n",
       "  11.8 ms\u001b[90m         Histogram: frequency by time\u001b[39m         13.9 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_bench = @benchmark c_sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C: Fastest time was 11.761558 msec\n"
     ]
    }
   ],
   "source": [
    "println(\"C: Fastest time was $(minimum(c_bench.times) / 1e6) msec\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 1 entry:\n",
       "  \"C\" => 11.7616"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = Dict()  # a \"dictionary\", i.e. an associative array\n",
    "d[\"C\"] = minimum(c_bench.times) / 1e6  # in milliseconds\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics # bring in statistical support for standard deviations\n",
    "t = c_bench.times / 1e6 # times in milliseconds\n",
    "m, σ = minimum(t), std(t)\n",
    "\n",
    "histogram(t, bins=500,\n",
    "    xlim=(m - 0.01, m + σ),\n",
    "    xlabel=\"milliseconds\", ylabel=\"count\", label=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. C with -ffast-math\n",
    "\n",
    "If we allow C to re-arrange the floating point operations, then it'll vectorize with SIMD (single instruction, multiple data) instructions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: redefinition of constant Clib_fastmath. This may fail, cause incorrect answers, or produce other errors.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "c_sum_fastmath (generic function with 1 method)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const Clib_fastmath = tempname()   # make a temporary file\n",
    "\n",
    "# The same as above but with a -ffast-math flag added\n",
    "open(`gcc -fPIC -O3 -msse3 -xc -shared -ffast-math -o $(Clib_fastmath * \".\" * Libdl.dlext) -`, \"w\") do f\n",
    "    print(f, C_code) \n",
    "end\n",
    "\n",
    "# define a Julia function that calls the C function:\n",
    "c_sum_fastmath(X::Array{Float64}) = ccall((\"c_sum\", Clib_fastmath), Float64, (Csize_t, Ptr{Float64}), length(X), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 1043 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.392 ms\u001b[22m\u001b[39m … \u001b[35m  7.165 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.630 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m4.780 ms\u001b[22m\u001b[39m ± \u001b[32m370.301 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m▄\u001b[39m█\u001b[39m▇\u001b[39m▃\u001b[39m▅\u001b[39m▂\u001b[39m \u001b[34m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[32m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▅\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[34m▅\u001b[39m\u001b[39m▅\u001b[39m▄\u001b[39m▄\u001b[39m▅\u001b[39m▆\u001b[32m▄\u001b[39m\u001b[39m▅\u001b[39m▇\u001b[39m▇\u001b[39m▆\u001b[39m▇\u001b[39m▅\u001b[39m▇\u001b[39m▅\u001b[39m▄\u001b[39m▄\u001b[39m▄\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▄\u001b[39m▂\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▃\u001b[39m▂\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m \u001b[39m▃\n",
       "  4.39 ms\u001b[90m         Histogram: frequency by time\u001b[39m        6.04 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_fastmath_bench = @benchmark $c_sum_fastmath($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.391986"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"C -ffast-math\"] = minimum(c_fastmath_bench.times) / 1e6  # in milliseconds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Python's built in `sum` "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `PyCall` package provides a Julia interface to Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "using Pkg; Pkg.add(\"PyCall\")\n",
    "using PyCall"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PyObject <built-in function sum>"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get the Python built-in \"sum\" function:\n",
    "pysum = pybuiltin(\"sum\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035265293e6"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pysum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pysum(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 8 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m700.439 ms\u001b[22m\u001b[39m … \u001b[35m715.469 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m702.104 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m705.163 ms\u001b[22m\u001b[39m ± \u001b[32m  6.090 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m▁\u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m█\u001b[34m \u001b[39m\u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[32m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \n",
       "  \u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m▁\u001b[39m█\u001b[34m▁\u001b[39m\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[32m▁\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m \u001b[39m▁\n",
       "  700 ms\u001b[90m           Histogram: frequency by time\u001b[39m          715 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m240 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py_list_bench = @benchmark $pysum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 3 entries:\n",
       "  \"C\"               => 11.7616\n",
       "  \"Python built-in\" => 700.439\n",
       "  \"C -ffast-math\"   => 4.39199"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python built-in\"] = minimum(py_list_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Python: `numpy` \n",
    "\n",
    "## Takes advantage of hardware \"SIMD\", but only works when it works.\n",
    "\n",
    "`numpy` is an optimized C library, callable from Python.\n",
    "It may be installed within Julia as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/github-personal/Introduction-to-Julia/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "using Pkg; Pkg.add(\"Conda\")\n",
    "using Conda"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mRunning `conda install -y numpy` in root environment\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Retrieving notices: ...working... done\n",
      "Collecting package metadata (current_repodata.json): ...working... done\n",
      "Solving environment: ...working... done\n",
      "\n",
      "## Package Plan ##\n",
      "\n",
      "  environment location: /Users/brobert/.julia/conda/3/x86_64\n",
      "\n",
      "  added / updated specs:\n",
      "    - numpy\n",
      "\n",
      "\n",
      "The following packages will be downloaded:\n",
      "\n",
      "    package                    |            build\n",
      "    ---------------------------|-----------------\n",
      "    conda-23.5.2               |  py310h2ec42d9_0        1009 KB  conda-forge\n",
      "    ------------------------------------------------------------\n",
      "                                           Total:        1009 KB\n",
      "\n",
      "The following packages will be UPDATED:\n",
      "\n",
      "  conda                              23.5.0-py310h2ec42d9_1 --> 23.5.2-py310h2ec42d9_0 \n",
      "\n",
      "\n",
      "\n",
      "Downloading and Extracting Packages\n",
      "                                                                               \n",
      "Preparing transaction: ...working... done\n",
      "Verifying transaction: ...working... done\n",
      "Executing transaction: ...working... done\n"
     ]
    }
   ],
   "source": [
    "Conda.add(\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 1112 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m3.950 ms\u001b[22m\u001b[39m … \u001b[35m  7.569 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.366 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m4.485 ms\u001b[22m\u001b[39m ± \u001b[32m446.676 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m▄\u001b[39m▃\u001b[39m█\u001b[39m▄\u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[34m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[32m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▇\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m▇\u001b[39m▆\u001b[39m▅\u001b[39m▅\u001b[39m▅\u001b[39m█\u001b[34m▆\u001b[39m\u001b[39m▅\u001b[39m▅\u001b[39m▆\u001b[32m▃\u001b[39m\u001b[39m▅\u001b[39m▄\u001b[39m▄\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▅\u001b[39m▅\u001b[39m▄\u001b[39m▆\u001b[39m▆\u001b[39m▅\u001b[39m▆\u001b[39m▆\u001b[39m▃\u001b[39m▅\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▃\n",
       "  3.95 ms\u001b[90m         Histogram: frequency by time\u001b[39m        5.78 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m240 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum = pyimport(\"numpy\")[\"sum\"]\n",
    "\n",
    "py_numpy_bench = @benchmark $numpy_sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035264871e6"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 4 entries:\n",
       "  \"C\"               => 11.7616\n",
       "  \"Python numpy\"    => 3.95018\n",
       "  \"Python built-in\" => 700.439\n",
       "  \"C -ffast-math\"   => 4.39199"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python numpy\"] = minimum(py_numpy_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5. Python, hand-written "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PyObject <function py_sum at 0x18a229d80>"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py\"\"\"\n",
    "def py_sum(A):\n",
    "    s = 0.0\n",
    "    for a in A:\n",
    "        s += a\n",
    "    return s\n",
    "\"\"\"\n",
    "\n",
    "sum_py = py\"py_sum\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 6 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m849.254 ms\u001b[22m\u001b[39m … \u001b[35m907.689 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m854.825 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m862.293 ms\u001b[22m\u001b[39m ± \u001b[32m 22.426 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m \u001b[39m▁\u001b[34m \u001b[39m\u001b[39m \u001b[39m█\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[32m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \n",
       "  \u001b[39m█\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[34m▁\u001b[39m\u001b[39m▁\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[32m▁\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m \u001b[39m▁\n",
       "  849 ms\u001b[90m           Histogram: frequency by time\u001b[39m          908 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m240 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py_hand = @benchmark $sum_py($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035265293e6"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_py(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_py(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 5 entries:\n",
       "  \"C\"                   => 11.7616\n",
       "  \"Python numpy\"        => 3.95018\n",
       "  \"Python hand-written\" => 849.254\n",
       "  \"Python built-in\"     => 700.439\n",
       "  \"C -ffast-math\"       => 4.39199"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python hand-written\"] = minimum(py_hand.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6. Julia (built-in) \n",
    "\n",
    "## Written directly in Julia, not in C!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "sum(a::<b>AbstractArray</b>; <i>dims, kw...</i>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/8e630552924eac54c809aa7bc30871c7df1582d3/base/reducedim.jl#L994\" target=\"_blank\">reducedim.jl:994</a>"
      ],
      "text/plain": [
       "sum(\u001b[90ma\u001b[39m::\u001b[1mAbstractArray\u001b[22m; dims, kw...)\n",
       "\u001b[90m     @\u001b[39m \u001b[90mBase\u001b[39m \u001b[90m\u001b[4mreducedim.jl:994\u001b[24m\u001b[39m"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 994 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.319 ms\u001b[22m\u001b[39m … \u001b[35m  7.035 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.968 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m5.013 ms\u001b[22m\u001b[39m ± \u001b[32m295.213 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m▅\u001b[39m▇\u001b[39m▅\u001b[39m█\u001b[39m▃\u001b[39m▃\u001b[34m▁\u001b[39m\u001b[39m▄\u001b[32m▁\u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▂\u001b[39m▂\u001b[39m▃\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▃\u001b[39m▂\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▂\u001b[39m▄\u001b[39m▄\u001b[39m▅\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[34m█\u001b[39m\u001b[39m█\u001b[32m█\u001b[39m\u001b[39m▇\u001b[39m▇\u001b[39m▃\u001b[39m▄\u001b[39m▅\u001b[39m▅\u001b[39m▅\u001b[39m▆\u001b[39m█\u001b[39m▇\u001b[39m▆\u001b[39m▆\u001b[39m▇\u001b[39m▄\u001b[39m▄\u001b[39m▄\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▃\u001b[39m▁\u001b[39m▃\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▃\n",
       "  4.32 ms\u001b[90m         Histogram: frequency by time\u001b[39m        5.74 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j_bench = @benchmark sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 6 entries:\n",
       "  \"C\"                   => 11.7616\n",
       "  \"Python numpy\"        => 3.95018\n",
       "  \"Python hand-written\" => 849.254\n",
       "  \"Python built-in\"     => 700.439\n",
       "  \"Julia built-in\"      => 4.31948\n",
       "  \"C -ffast-math\"       => 4.39199"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia built-in\"] = minimum(j_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Julia (hand-written) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mysum (generic function with 1 method)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mysum(A)   \n",
    "    s = 0.0 # s = zero(eltype(a))\n",
    "    for a in A\n",
    "        s += a\n",
    "    end\n",
    "    s\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 396 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m12.139 ms\u001b[22m\u001b[39m … \u001b[35m 14.543 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m12.627 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m12.638 ms\u001b[22m\u001b[39m ± \u001b[32m306.080 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m \u001b[39m▃\u001b[39m \u001b[39m \u001b[39m▂\u001b[39m▄\u001b[39m▂\u001b[39m▆\u001b[39m \u001b[39m \u001b[39m \u001b[39m▆\u001b[39m \u001b[39m \u001b[39m▄\u001b[39m▄\u001b[39m▃\u001b[39m█\u001b[34m \u001b[39m\u001b[39m▄\u001b[39m▂\u001b[39m▄\u001b[39m▄\u001b[39m▇\u001b[39m▂\u001b[39m▄\u001b[39m▄\u001b[39m▄\u001b[39m▁\u001b[39m▂\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▅\u001b[39m▆\u001b[39m█\u001b[39m█\u001b[39m▆\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m▇\u001b[39m█\u001b[39m▆\u001b[39m█\u001b[39m▆\u001b[39m▇\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[34m▇\u001b[39m\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m▄\u001b[39m▆\u001b[39m▅\u001b[39m▁\u001b[39m▁\u001b[39m▄\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▁\u001b[39m▁\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▁\u001b[39m▃\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▃\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▃\u001b[39m \u001b[39m▄\n",
       "  12.1 ms\u001b[90m         Histogram: frequency by time\u001b[39m         13.7 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j_bench_hand = @benchmark mysum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 7 entries:\n",
       "  \"C\"                   => 11.7616\n",
       "  \"Python numpy\"        => 3.95018\n",
       "  \"Julia hand-written\"  => 12.1394\n",
       "  \"Python hand-written\" => 849.254\n",
       "  \"Python built-in\"     => 700.439\n",
       "  \"Julia built-in\"      => 4.31948\n",
       "  \"C -ffast-math\"       => 4.39199"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia hand-written\"] = minimum(j_bench_hand.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8. Julia (hand-written w. simd) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mysum_simd (generic function with 1 method)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mysum_simd(A)   \n",
    "    s = 0.0 # s = zero(eltype(A))\n",
    "    @simd for a in A\n",
    "        s += a\n",
    "    end\n",
    "    s\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 956 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.461 ms\u001b[22m\u001b[39m … \u001b[35m  6.894 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m5.152 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m5.213 ms\u001b[22m\u001b[39m ± \u001b[32m206.971 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▂\u001b[39m▂\u001b[39m▄\u001b[39m▅\u001b[39m▅\u001b[39m▄\u001b[39m▆\u001b[39m█\u001b[34m▁\u001b[39m\u001b[39m▁\u001b[39m \u001b[32m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m▁\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▁\u001b[39m▁\u001b[39m▂\u001b[39m▂\u001b[39m▁\u001b[39m▂\u001b[39m▃\u001b[39m▁\u001b[39m▃\u001b[39m▂\u001b[39m▃\u001b[39m▄\u001b[39m▇\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[39m█\u001b[34m█\u001b[39m\u001b[39m█\u001b[39m▅\u001b[32m▆\u001b[39m\u001b[39m▇\u001b[39m▆\u001b[39m▆\u001b[39m▇\u001b[39m▅\u001b[39m▇\u001b[39m█\u001b[39m█\u001b[39m▇\u001b[39m▇\u001b[39m▇\u001b[39m▅\u001b[39m▅\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▄\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m▃\u001b[39m \u001b[39m▄\n",
       "  4.46 ms\u001b[90m         Histogram: frequency by time\u001b[39m        5.74 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j_bench_hand_simd = @benchmark mysum_simd($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999754035264923e6"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysum_simd(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 8 entries:\n",
       "  \"Julia hand-written simd\" => 4.46098\n",
       "  \"C\"                       => 11.7616\n",
       "  \"Python numpy\"            => 3.95018\n",
       "  \"Julia hand-written\"      => 12.1394\n",
       "  \"Python hand-written\"     => 849.254\n",
       "  \"Python built-in\"         => 700.439\n",
       "  \"Julia built-in\"          => 4.31948\n",
       "  \"C -ffast-math\"           => 4.39199"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia hand-written simd\"] = minimum(j_bench_hand_simd.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Summary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python numpy................4.0\n",
      "Julia built-in..............4.3\n",
      "C -ffast-math...............4.4\n",
      "Julia hand-written simd.....4.5\n",
      "C..........................11.8\n",
      "Julia hand-written.........12.1\n",
      "Python built-in...........700.4\n",
      "Python hand-written.......849.3\n"
     ]
    }
   ],
   "source": [
    "for (key, value) in sort(collect(d), by=last)\n",
    "    println(rpad(key, 25, \".\"), lpad(round(value; digits=1), 6, \".\"))\n",
    "end"
   ]
  }
 ],
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