solidity/test/compilationTests/gnosis/Utils/Math.sol
2019-11-14 11:49:39 +01:00

342 lines
9.7 KiB
Solidity

pragma solidity >=0.0;
/// @title Math library - Allows calculation of logarithmic and exponential functions
/// @author Alan Lu - <alan.lu@gnosis.pm>
/// @author Stefan George - <stefan@gnosis.pm>
library Math {
/*
* Constants
*/
// This is equal to 1 in our calculations
uint public constant ONE = 0x10000000000000000;
uint public constant LN2 = 0xb17217f7d1cf79ac;
uint public constant LOG2_E = 0x171547652b82fe177;
/*
* Public functions
*/
/// @dev Returns natural exponential function value of given x
/// @param x x
/// @return e**x
function exp(int x)
public
pure
returns (uint)
{
// revert if x is > MAX_POWER, where
// MAX_POWER = int(mp.floor(mp.log(mpf(2**256 - 1) / ONE) * ONE))
require(x <= 2454971259878909886679);
// return 0 if exp(x) is tiny, using
// MIN_POWER = int(mp.floor(mp.log(mpf(1) / ONE) * ONE))
if (x < -818323753292969962227)
return 0;
// Transform so that e^x -> 2^x
x = x * int(ONE) / int(LN2);
// 2^x = 2^whole(x) * 2^frac(x)
// ^^^^^^^^^^ is a bit shift
// so Taylor expand on z = frac(x)
int shift;
uint z;
if (x >= 0) {
shift = x / int(ONE);
z = uint(x % int(ONE));
}
else {
shift = x / int(ONE) - 1;
z = ONE - uint(-x % int(ONE));
}
// 2^x = 1 + (ln 2) x + (ln 2)^2/2! x^2 + ...
//
// Can generate the z coefficients using mpmath and the following lines
// >>> from mpmath import mp
// >>> mp.dps = 100
// >>> ONE = 0x10000000000000000
// >>> print('\n'.join(hex(int(mp.log(2)**i / mp.factorial(i) * ONE)) for i in range(1, 7)))
// 0xb17217f7d1cf79ab
// 0x3d7f7bff058b1d50
// 0xe35846b82505fc5
// 0x276556df749cee5
// 0x5761ff9e299cc4
// 0xa184897c363c3
uint zpow = z;
uint result = ONE;
result += 0xb17217f7d1cf79ab * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x3d7f7bff058b1d50 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0xe35846b82505fc5 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x276556df749cee5 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x5761ff9e299cc4 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0xa184897c363c3 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0xffe5fe2c4586 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x162c0223a5c8 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x1b5253d395e * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x1e4cf5158b * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x1e8cac735 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x1c3bd650 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x1816193 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x131496 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0xe1b7 * zpow / ONE;
zpow = zpow * z / ONE;
result += 0x9c7 * zpow / ONE;
if (shift >= 0) {
if (result >> (256-shift) > 0)
return (2**256-1);
return result << shift;
}
else
return result >> (-shift);
}
/// @dev Returns natural logarithm value of given x
/// @param x x
/// @return ln(x)
function ln(uint x)
public
pure
returns (int)
{
require(x > 0);
// binary search for floor(log2(x))
int ilog2 = floorLog2(x);
int z;
if (ilog2 < 0)
z = int(x << uint(-ilog2));
else
z = int(x >> uint(ilog2));
// z = x * 2^-⌊log₂x⌋
// so 1 <= z < 2
// and ln z = ln x - ⌊log₂x⌋/log₂e
// so just compute ln z using artanh series
// and calculate ln x from that
int term = (z - int(ONE)) * int(ONE) / (z + int(ONE));
int halflnz = term;
int termpow = term * term / int(ONE) * term / int(ONE);
halflnz += termpow / 3;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 5;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 7;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 9;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 11;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 13;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 15;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 17;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 19;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 21;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 23;
termpow = termpow * term / int(ONE) * term / int(ONE);
halflnz += termpow / 25;
return (ilog2 * int(ONE)) * int(ONE) / int(LOG2_E) + 2 * halflnz;
}
/// @dev Returns base 2 logarithm value of given x
/// @param x x
/// @return lo logarithmic value
function floorLog2(uint x)
public
pure
returns (int lo)
{
lo = -64;
int hi = 193;
// I use a shift here instead of / 2 because it floors instead of rounding towards 0
int mid = (hi + lo) >> 1;
while((lo + 1) < hi) {
if (mid < 0 && x << uint(-mid) < ONE || mid >= 0 && x >> uint(mid) < ONE)
hi = mid;
else
lo = mid;
mid = (hi + lo) >> 1;
}
}
/// @dev Returns maximum of an array
/// @param nums Numbers to look through
/// @return max Maximum number
function max(int[] memory nums)
virtual
public
pure
returns (int max)
{
require(nums.length > 0);
max = -2**255;
for (uint i = 0; i < nums.length; i++)
if (nums[i] > max)
max = nums[i];
}
/// @dev Returns whether an add operation causes an overflow
/// @param a First addend
/// @param b Second addend
/// @return Did no overflow occur?
function safeToAdd(uint a, uint b)
public
pure
returns (bool)
{
return a + b >= a;
}
/// @dev Returns whether a subtraction operation causes an underflow
/// @param a Minuend
/// @param b Subtrahend
/// @return Did no underflow occur?
function safeToSub(uint a, uint b)
public
pure
returns (bool)
{
return a >= b;
}
/// @dev Returns whether a multiply operation causes an overflow
/// @param a First factor
/// @param b Second factor
/// @return Did no overflow occur?
function safeToMul(uint a, uint b)
public
pure
returns (bool)
{
return b == 0 || a * b / b == a;
}
/// @dev Returns sum if no overflow occurred
/// @param a First addend
/// @param b Second addend
/// @return Sum
function add(uint a, uint b)
public
pure
returns (uint)
{
require(safeToAdd(a, b));
return a + b;
}
/// @dev Returns difference if no overflow occurred
/// @param a Minuend
/// @param b Subtrahend
/// @return Difference
function sub(uint a, uint b)
public
pure
returns (uint)
{
require(safeToSub(a, b));
return a - b;
}
/// @dev Returns product if no overflow occurred
/// @param a First factor
/// @param b Second factor
/// @return Product
function mul(uint a, uint b)
public
pure
returns (uint)
{
require(safeToMul(a, b));
return a * b;
}
/// @dev Returns whether an add operation causes an overflow
/// @param a First addend
/// @param b Second addend
/// @return Did no overflow occur?
function safeToAdd(int a, int b)
public
pure
returns (bool)
{
return (b >= 0 && a + b >= a) || (b < 0 && a + b < a);
}
/// @dev Returns whether a subtraction operation causes an underflow
/// @param a Minuend
/// @param b Subtrahend
/// @return Did no underflow occur?
function safeToSub(int a, int b)
public
pure
returns (bool)
{
return (b >= 0 && a - b <= a) || (b < 0 && a - b > a);
}
/// @dev Returns whether a multiply operation causes an overflow
/// @param a First factor
/// @param b Second factor
/// @return Did no overflow occur?
function safeToMul(int a, int b)
public
pure
returns (bool)
{
return (b == 0) || (a * b / b == a);
}
/// @dev Returns sum if no overflow occurred
/// @param a First addend
/// @param b Second addend
/// @return Sum
function add(int a, int b)
public
pure
returns (int)
{
require(safeToAdd(a, b));
return a + b;
}
/// @dev Returns difference if no overflow occurred
/// @param a Minuend
/// @param b Subtrahend
/// @return Difference
function sub(int a, int b)
public
pure
returns (int)
{
require(safeToSub(a, b));
return a - b;
}
/// @dev Returns product if no overflow occurred
/// @param a First factor
/// @param b Second factor
/// @return Product
function mul(int a, int b)
public
pure
returns (int)
{
require(safeToMul(a, b));
return a * b;
}
}