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AdvancedMath.py
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from .common.BuiltIn import *
from . import AnalyticMath
from . import FractionMath
from . import IntegralMath
FIXED_1 = AnalyticMath.FIXED_1;
# Auto-generated via 'PrintAdvancedMathConstants.py'
LAMBERT_NEG1_MAXVAL = 0x002e9e207581ee21a2fdc7e0328c907634;
LAMBERT_NEG2_MAXVAL = 0x002f16ac6c59de6f8d5d6f63c1482a7c86;
LAMBERT_NEG2_SAMPLE = 0x0000080954b9100531c21c3bf872e8228e;
LAMBERT_NEG2_T_SIZE = 0x0000000000000000000000000000000011;
LAMBERT_NEG2_T_MASK = 0xffffffffffffffffffffffffffffffffff;
LAMBERT_POS1_MAXVAL = 0x002f16ac6c59de6f8d5d6f63c1482a7c86;
LAMBERT_POS2_MAXVAL = 0x0c2f16ac6c59de6f8d5d6f63c1482a7c56;
LAMBERT_POS2_SAMPLE = 0x00183060c183060c183060c183060c1830;
LAMBERT_POS2_T_SIZE = 0x0000000000000000000000000000000010;
LAMBERT_POS2_T_MASK = 0x00ffffffffffffffffffffffffffffffff;
LAMBERT_NEG2_VALUES = "012ff3f5b94a776318e70a4b0be31da9a5"\
"01314a4702ae8ae09a01a03f6933dcf01f"\
"0132afb4b8985b32518cd371f2b5c6de56"\
"0134260eac628aa55b0b35bd5ba1df0e6e"\
"0135af881e5cbc17f5de16bb05816826c0"\
"01374ed8520d0404e91fe8e683b8d8173e"\
"0139076a4780adf3619a57f1b6e40e3f20"\
"013adda534e8e43471fe0ae32b8bbf0cfc"\
"013cd75f834551aa105eff810be8afd15e"\
"013efc9f6e12358e1f8c040050efadd3e2"\
"014158f4e68e9217def733dc63fba7eba9"\
"0143fe22b87c9944205ebc3be59f410691"\
"01470a158303eabda73222305243e7c7b2"\
"014ab733a071f0dfc80ab6c42b29b5ce07"\
"014f9c668b59483575a0daad8006952948"\
"015aa59d425cbd5ed5a6a64da5a1108555";
LAMBERT_POS2_VALUES = "60e393c68d20b1bd09deaabc0373b9c5"\
"577b97aa1fe222bb452fdf111b1f0be2"\
"5035f241d6eae0cd7bacba119993de7b"\
"4a5cbc96a05589cb4d86be1db3168364"\
"4585c8b3f8fe489c6e1833ca47871384"\
"416e1b785d13eba07a08f3f18876a5ab"\
"3de8f65ac388101ddf718a6f5c1eff65"\
"3ad71c1c77e34fa32a9f184967eccbf6"\
"3821f57dbd2763256c1a99bbd2051378"\
"35b8b28d1a73dc27500ffe35559cc028"\
"338e82ce00e2496262c64457535ba1a1"\
"31996ec6b07b4a83421b5ebc4ab4e1f1"\
"2fd19346ed17dac61219ce0c2c5ac4b0"\
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"2a07c206cde2c9467e42ebdfc395e285"\
"28d6e752efcbae881bb2163e088236de"\
"27ba81d008f57136440e87f44fb95f59"\
"26b06bf4b3ca8a8e6b5f1207cf63e484"\
"25b6cd7b40b306b014cc93eb0fd34b1b"\
"24cc0df118d221d961d0e3c7c2cc74d8"\
"23eeca079ec9f001c7488cb0210969d3"\
"231dcb016e2797ddce1ba75a69081655"\
"2257ffc1f84048f54f9072007f255977"\
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"20ea5b53c3a2de83f69655808913322d"\
"2040edf2ba98675afaf33ad7b36a7956"\
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"0fc4c7714e3aeceb64a83c0fbd5b3409"\
"0fa6aaf54861476764473f34e82d3a9c"\
"0f8912b528ab0b6a790407fa5f9bf8b1"\
"0f6bfb1fa7349e64712ab860554d7b78"\
"0f4f60c509968df5add12deae5027309"\
"0f334055948b2a0dc667a844c6723b23"\
"0f1796a013d5b15214dc09abf2a0592b"\
"0efc609076f78738f15c360223f8c79d"\
"0ee19b2e815b8e957ce06c5e16449ca8"\
"0ec7439c8cb9187b324eb1d2758ee4d3"\
"0ead57165c8714db919b9ccd15e45159"\
"0e93d2f0016d88ff1e5182897a357740"\
"0e7ab494cbb6ec0c4cbc602eb4bb78e3"\
"0e61f9864bd512e429ac81729beeec69"\
"0e499f5b601dee819305eb00790b9c9a"\
"0e31a3bf4ef3ab0dadc11b18d962c820"\
"0e1a0470ec99d736c6436de2abaf3b64"\
"0e02bf41cc063d700f4cb7817df3dfd7"\
"0debd2157a081b83117c46ea1dc579d1"\
"0dd53ae0c22b7df344fa51532b60dd98"\
"0dbef7a8fcc8c7c9c5332ac339c9f88d"\
"0da9068365b9ee4b304fb9666adccf46"\
"0d9365947b37bb14237ee7d1802ffa2f"\
"0d7e130f64698ce1098c56ddad89fb56"\
"0d690d355f399d48cad4f9225f1a3f9d"\
"0d5452553506d42a1f2f7d5cf97679d9"\
"0d3fe0cab5d3b39f329597f217bf7928"\
"0d2bb6fe3997f32783f2b5d8a3acd0c9"\
"0d17d364275fffa6cb83e0553ca42663"\
"0d04347c81ead056e793c81ed9aa583e"\
"0cf0d8d2797b5e9e40afe04977971655"\
"0cddbefc029796ff51a1c94f6c2e5589"\
"0ccae5997172d28b9d7969676b6cfb06"\
"0cb84b5519c5d86400e5ebbcb895c475"\
"0ca5eee2f2da13533492e3978c84b997"\
"0c93cf003f91157bbee2579fcf0a33f3"\
"0c81ea733a34b805f2bf6edd83fca2cd"\
"0c70400ac3df22f2e73d2a54bee3b3ee"\
"0c5ece9e174cd59d6c32f900259dd921"\
"0c4d950c7eed66993aa3f25a85ea9e43";
LAMBERT_EXACT_LIMIT = 0x001abb58194f861a372fc4a5e2a64806c77ee7849dd0a347ea7ddf2ba582711b;
'''
@dev Solve x * (a / b) ^ x = c / d
Solution: x = W(log(a / b) * c / d) / (log(a / b) * c / d) * c / d
'''
def solveExact(a, b, c, d):
return solve(a, b, c, d, lambertNegExact, lambertPosExact);
'''
@dev Solve x * (a / b) ^ x = c / d
Solution: x = W(log(a / b) * c / d) / (log(a / b) * c / d) * c / d
'''
def solveQuick(a, b, c, d):
return solve(a, b, c, d, lambertNegQuick, lambertPosQuick);
'''
@dev Compute W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
Input range: 1 <= x <= LAMBERT_NEG2_MAXVAL
'''
def lambertNegExact(x):
require(x > 0, "lambertNegExact: x < min");
if (x <= LAMBERT_NEG2_MAXVAL):
y = x;
for i in range(8):
e = AnalyticMath.fixedExp(y);
y = (x * e - y * y) // (FIXED_1 - y);
return IntegralMath.mulDivF(FIXED_1, y, x);
revert("lambertNegExact: x > max");
'''
@dev Compute W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
Input range: 1 <= x <= 2 ^ 256 - 1
'''
def lambertPosExact(x):
require(x > 0, "lambertPosExact: x < min");
if (x <= LAMBERT_EXACT_LIMIT):
if (x < FIXED_1):
p = AnalyticMath.fixedExp(x);
q = IntegralMath.mulDivF(x, p, FIXED_1);
y = IntegralMath.mulDivF(x, q + FIXED_1, q + p);
else:
p = AnalyticMath.fixedLog(x);
q = IntegralMath.mulDivF(p, p, FIXED_1);
y = IntegralMath.mulDivF(FIXED_1, q + FIXED_1, p + FIXED_1);
for i in range(7):
e = AnalyticMath.fixedExp(y);
f = IntegralMath.mulDivF(y, e, FIXED_1);
g = IntegralMath.mulDivF(y, f, FIXED_1);
y = IntegralMath.mulDivF(FIXED_1, g + x, f + e);
return IntegralMath.mulDivF(FIXED_1, y, x);
return lambertPos3(x);
'''
@dev Compute W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
Input range: 1 <= x <= LAMBERT_NEG2_MAXVAL
'''
def lambertNegQuick(x):
require(x > 0, "lambertNegQuick: x < min");
if (x <= LAMBERT_NEG1_MAXVAL):
return lambertNeg1(x);
if (x <= LAMBERT_NEG2_MAXVAL):
return lambertNeg2(x);
revert("lambertNegQuick: x > max");
'''
@dev Compute W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
Input range: 1 <= x <= 2 ^ 256 - 1
'''
def lambertPosQuick(x):
require(x > 0, "lambertPosQuick: x < min");
if (x <= LAMBERT_POS1_MAXVAL):
return lambertPos1(x);
if (x <= LAMBERT_POS2_MAXVAL):
return lambertPos2(x);
return lambertPos3(x);
'''
@dev Compute W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
Input range: 1 <= x <= LAMBERT_NEG1_MAXVAL
Auto-generated via 'PrintAdvancedMathLambertNeg1.py'
'''
def lambertNeg1(x):
res = 0;
xi = x;
xi = (xi * x) // FIXED_1; res += xi * 0x00000000014d29a73a6e7b02c3668c7b0880000000; # add x^(03-1) * (34! * 03^(03-1) / 03!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000002504a0cd9a7f7215b60f9be4800000000; # add x^(04-1) * (34! * 04^(04-1) / 04!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000000484d0a1191c0ead267967c7a4a0000000; # add x^(05-1) * (34! * 05^(05-1) / 05!)
xi = (xi * x) // FIXED_1; res += xi * 0x00000000095ec580d7e8427a4baf26a90a00000000; # add x^(06-1) * (34! * 06^(06-1) / 06!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000001440b0be1615a47dba6e5b3b1f10000000; # add x^(07-1) * (34! * 07^(07-1) / 07!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000002d207601f46a99b4112418400000000000; # add x^(08-1) * (34! * 08^(08-1) / 08!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000066ebaac4c37c622dd8288a7eb1b2000000; # add x^(09-1) * (34! * 09^(09-1) / 09!)
xi = (xi * x) // FIXED_1; res += xi * 0x00000000ef17240135f7dbd43a1ba10cf200000000; # add x^(10-1) * (34! * 10^(10-1) / 10!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000233c33c676a5eb2416094a87b3657000000; # add x^(11-1) * (34! * 11^(11-1) / 11!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000541cde48bc0254bed49a9f8700000000000; # add x^(12-1) * (34! * 12^(12-1) / 12!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000cae1fad2cdd4d4cb8d73abca0d19a400000; # add x^(13-1) * (34! * 13^(13-1) / 13!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000001edb2aa2f760d15c41ceedba956400000000; # add x^(14-1) * (34! * 14^(14-1) / 14!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000004ba8d20d2dabd386c9529659841a2e200000; # add x^(15-1) * (34! * 15^(15-1) / 15!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000bac08546b867cdaa20000000000000000000; # add x^(16-1) * (34! * 16^(16-1) / 16!)
xi = (xi * x) // FIXED_1; res += xi * 0x000001cfa8e70c03625b9db76c8ebf5bbf24820000; # add x^(17-1) * (34! * 17^(17-1) / 17!)
xi = (xi * x) // FIXED_1; res += xi * 0x000004851d99f82060df265f3309b26f8200000000; # add x^(18-1) * (34! * 18^(18-1) / 18!)
xi = (xi * x) // FIXED_1; res += xi * 0x00000b550d19b129d270c44f6f55f027723cbb0000; # add x^(19-1) * (34! * 19^(19-1) / 19!)
xi = (xi * x) // FIXED_1; res += xi * 0x00001c877dadc761dc272deb65d4b0000000000000; # add x^(20-1) * (34! * 20^(20-1) / 20!)
xi = (xi * x) // FIXED_1; res += xi * 0x000048178ece97479f33a77f2ad22a81b64406c000; # add x^(21-1) * (34! * 21^(21-1) / 21!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000b6ca8268b9d810fedf6695ef2f8a6c00000000; # add x^(22-1) * (34! * 22^(22-1) / 22!)
xi = (xi * x) // FIXED_1; res += xi * 0x0001d0e76631a5b05d007b8cb72a7c7f11ec36e000; # add x^(23-1) * (34! * 23^(23-1) / 23!)
xi = (xi * x) // FIXED_1; res += xi * 0x0004a1c37bd9f85fd9c6c780000000000000000000; # add x^(24-1) * (34! * 24^(24-1) / 24!)
xi = (xi * x) // FIXED_1; res += xi * 0x000bd8369f1b702bf491e2ebfcee08250313b65400; # add x^(25-1) * (34! * 25^(25-1) / 25!)
xi = (xi * x) // FIXED_1; res += xi * 0x001e5c7c32a9f6c70ab2cb59d9225764d400000000; # add x^(26-1) * (34! * 26^(26-1) / 26!)
xi = (xi * x) // FIXED_1; res += xi * 0x004dff5820e165e910f95120a708e742496221e600; # add x^(27-1) * (34! * 27^(27-1) / 27!)
xi = (xi * x) // FIXED_1; res += xi * 0x00c8c8f66db1fced378ee50e536000000000000000; # add x^(28-1) * (34! * 28^(28-1) / 28!)
xi = (xi * x) // FIXED_1; res += xi * 0x0205db8dffff45bfa2938f128f599dbf16eb11d880; # add x^(29-1) * (34! * 29^(29-1) / 29!)
xi = (xi * x) // FIXED_1; res += xi * 0x053a044ebd984351493e1786af38d39a0800000000; # add x^(30-1) * (34! * 30^(30-1) / 30!)
xi = (xi * x) // FIXED_1; res += xi * 0x0d86dae2a4cc0f47633a544479735869b487b59c40; # add x^(31-1) * (34! * 31^(31-1) / 31!)
xi = (xi * x) // FIXED_1; res += xi * 0x231000000000000000000000000000000000000000; # add x^(32-1) * (34! * 32^(32-1) / 32!)
xi = (xi * x) // FIXED_1; res += xi * 0x5b0485a76f6646c2039db1507cdd51b08649680822; # add x^(33-1) * (34! * 33^(33-1) / 33!)
xi = (xi * x) // FIXED_1; res += xi * 0xec983c46c49545bc17efa6b5b0055e242200000000; # add x^(34-1) * (34! * 34^(34-1) / 34!)
return res // 0xde1bc4d19efcac82445da75b00000000 + FIXED_1 + x; # divide by 34! and then add x^(1-1) * (1^(1-1) / 1!) + x^(2-1) * (2^(2-1) / 2!)
'''
@dev Compute W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
Input range: 1 <= x <= LAMBERT_POS1_MAXVAL
Auto-generated via 'PrintAdvancedMathLambertPos1.py'
'''
def lambertPos1(x):
res = 0;
xi = x;
xi = (xi * x) // FIXED_1; res += xi * 0x00000000014d29a73a6e7b02c3668c7b0880000000; # add x^(03-1) * (34! * 03^(03-1) / 03!)
xi = (xi * x) // FIXED_1; res -= xi * 0x0000000002504a0cd9a7f7215b60f9be4800000000; # sub x^(04-1) * (34! * 04^(04-1) / 04!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000000484d0a1191c0ead267967c7a4a0000000; # add x^(05-1) * (34! * 05^(05-1) / 05!)
xi = (xi * x) // FIXED_1; res -= xi * 0x00000000095ec580d7e8427a4baf26a90a00000000; # sub x^(06-1) * (34! * 06^(06-1) / 06!)
xi = (xi * x) // FIXED_1; res += xi * 0x000000001440b0be1615a47dba6e5b3b1f10000000; # add x^(07-1) * (34! * 07^(07-1) / 07!)
xi = (xi * x) // FIXED_1; res -= xi * 0x000000002d207601f46a99b4112418400000000000; # sub x^(08-1) * (34! * 08^(08-1) / 08!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000066ebaac4c37c622dd8288a7eb1b2000000; # add x^(09-1) * (34! * 09^(09-1) / 09!)
xi = (xi * x) // FIXED_1; res -= xi * 0x00000000ef17240135f7dbd43a1ba10cf200000000; # sub x^(10-1) * (34! * 10^(10-1) / 10!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000233c33c676a5eb2416094a87b3657000000; # add x^(11-1) * (34! * 11^(11-1) / 11!)
xi = (xi * x) // FIXED_1; res -= xi * 0x0000000541cde48bc0254bed49a9f8700000000000; # sub x^(12-1) * (34! * 12^(12-1) / 12!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000000cae1fad2cdd4d4cb8d73abca0d19a400000; # add x^(13-1) * (34! * 13^(13-1) / 13!)
xi = (xi * x) // FIXED_1; res -= xi * 0x0000001edb2aa2f760d15c41ceedba956400000000; # sub x^(14-1) * (34! * 14^(14-1) / 14!)
xi = (xi * x) // FIXED_1; res += xi * 0x0000004ba8d20d2dabd386c9529659841a2e200000; # add x^(15-1) * (34! * 15^(15-1) / 15!)
xi = (xi * x) // FIXED_1; res -= xi * 0x000000bac08546b867cdaa20000000000000000000; # sub x^(16-1) * (34! * 16^(16-1) / 16!)
xi = (xi * x) // FIXED_1; res += xi * 0x000001cfa8e70c03625b9db76c8ebf5bbf24820000; # add x^(17-1) * (34! * 17^(17-1) / 17!)
xi = (xi * x) // FIXED_1; res -= xi * 0x000004851d99f82060df265f3309b26f8200000000; # sub x^(18-1) * (34! * 18^(18-1) / 18!)
xi = (xi * x) // FIXED_1; res += xi * 0x00000b550d19b129d270c44f6f55f027723cbb0000; # add x^(19-1) * (34! * 19^(19-1) / 19!)
xi = (xi * x) // FIXED_1; res -= xi * 0x00001c877dadc761dc272deb65d4b0000000000000; # sub x^(20-1) * (34! * 20^(20-1) / 20!)
xi = (xi * x) // FIXED_1; res += xi * 0x000048178ece97479f33a77f2ad22a81b64406c000; # add x^(21-1) * (34! * 21^(21-1) / 21!)
xi = (xi * x) // FIXED_1; res -= xi * 0x0000b6ca8268b9d810fedf6695ef2f8a6c00000000; # sub x^(22-1) * (34! * 22^(22-1) / 22!)
xi = (xi * x) // FIXED_1; res += xi * 0x0001d0e76631a5b05d007b8cb72a7c7f11ec36e000; # add x^(23-1) * (34! * 23^(23-1) / 23!)
xi = (xi * x) // FIXED_1; res -= xi * 0x0004a1c37bd9f85fd9c6c780000000000000000000; # sub x^(24-1) * (34! * 24^(24-1) / 24!)
xi = (xi * x) // FIXED_1; res += xi * 0x000bd8369f1b702bf491e2ebfcee08250313b65400; # add x^(25-1) * (34! * 25^(25-1) / 25!)
xi = (xi * x) // FIXED_1; res -= xi * 0x001e5c7c32a9f6c70ab2cb59d9225764d400000000; # sub x^(26-1) * (34! * 26^(26-1) / 26!)
xi = (xi * x) // FIXED_1; res += xi * 0x004dff5820e165e910f95120a708e742496221e600; # add x^(27-1) * (34! * 27^(27-1) / 27!)
xi = (xi * x) // FIXED_1; res -= xi * 0x00c8c8f66db1fced378ee50e536000000000000000; # sub x^(28-1) * (34! * 28^(28-1) / 28!)
xi = (xi * x) // FIXED_1; res += xi * 0x0205db8dffff45bfa2938f128f599dbf16eb11d880; # add x^(29-1) * (34! * 29^(29-1) / 29!)
xi = (xi * x) // FIXED_1; res -= xi * 0x053a044ebd984351493e1786af38d39a0800000000; # sub x^(30-1) * (34! * 30^(30-1) / 30!)
xi = (xi * x) // FIXED_1; res += xi * 0x0d86dae2a4cc0f47633a544479735869b487b59c40; # add x^(31-1) * (34! * 31^(31-1) / 31!)
xi = (xi * x) // FIXED_1; res -= xi * 0x231000000000000000000000000000000000000000; # sub x^(32-1) * (34! * 32^(32-1) / 32!)
xi = (xi * x) // FIXED_1; res += xi * 0x5b0485a76f6646c2039db1507cdd51b08649680822; # add x^(33-1) * (34! * 33^(33-1) / 33!)
xi = (xi * x) // FIXED_1; res -= xi * 0xec983c46c49545bc17efa6b5b0055e242200000000; # sub x^(34-1) * (34! * 34^(34-1) / 34!)
return res // 0xde1bc4d19efcac82445da75b00000000 + FIXED_1 - x; # divide by 34! and then add x^(1-1) * (1^(1-1) / 1!) - x^(2-1) * (2^(2-1) / 2!)
'''
@dev Compute W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
Input range: LAMBERT_NEG1_MAXVAL + 1 <= x <= LAMBERT_NEG2_MAXVAL
'''
def lambertNeg2(x):
values = LAMBERT_NEG2_VALUES;
y = x - LAMBERT_NEG1_MAXVAL - 1;
i = y // LAMBERT_NEG2_SAMPLE;
a = LAMBERT_NEG2_SAMPLE * (i + 0);
b = LAMBERT_NEG2_SAMPLE * (i + 1);
c = LAMBERT_NEG2_T_SIZE * (i + 1);
d = LAMBERT_NEG2_T_SIZE * (i + 2);
e = read(values, c) & LAMBERT_NEG2_T_MASK;
f = read(values, d) & LAMBERT_NEG2_T_MASK;
g = e * (b - y);
h = f * (y - a);
return (g + h) // LAMBERT_NEG2_SAMPLE;
'''
@dev Compute W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
Input range: LAMBERT_POS1_MAXVAL + 1 <= x <= LAMBERT_POS2_MAXVAL
'''
def lambertPos2(x):
values = LAMBERT_POS2_VALUES;
y = x - LAMBERT_POS1_MAXVAL - 1;
i = y // LAMBERT_POS2_SAMPLE;
a = LAMBERT_POS2_SAMPLE * (i + 0);
b = LAMBERT_POS2_SAMPLE * (i + 1);
c = LAMBERT_POS2_T_SIZE * (i + 1);
d = LAMBERT_POS2_T_SIZE * (i + 2);
e = read(values, c) & LAMBERT_POS2_T_MASK;
f = read(values, d) & LAMBERT_POS2_T_MASK;
g = e * (b - y);
h = f * (y - a);
return (g + h) // LAMBERT_POS2_SAMPLE;
'''
@dev Compute W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
Input range: LAMBERT_POS2_MAXVAL + 1 <= x <= 2 ^ 256 - 1
'''
def lambertPos3(x):
a = AnalyticMath.fixedLog(x);
b = AnalyticMath.fixedLog(a);
c = IntegralMath.mulDivF(a, a, FIXED_1 // 2);
d = IntegralMath.mulDivF(b, b + (a - FIXED_1) * 2, c);
e = IntegralMath.mulDivF(FIXED_1, a - b + d, x);
return e;
'''
@dev Solve the equation x * (a / b) ^ x = c / d for x
'''
def solve(a, b, c, d, lambertNeg, lambertPos):
if (a > b):
return call(lambertPos, a, b, c, d);
if (b > a):
return call(lambertNeg, b, a, c, d);
return (c, d);
'''
@dev Return f(log(x / y) * z / w * FIXED_1) * z / w / FIXED_1
'''
def call(f, x, y, z, w):
return FractionMath.productRatio(
f(
IntegralMath.mulDivF(
AnalyticMath.fixedLog(
IntegralMath.mulDivF(
FIXED_1, x, y
)
), z, w
)
), z, w, FIXED_1
);
'''
@dev Read 32 bytes from a given chunk of data at a given offset
'''
def read(data, offset):
return int(("0" * 64 + data)[offset * 2 : offset * 2 + 64], 16);