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lib/abdk-math/ABDKMath64x64.md

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+ABDK Math 64.64
+===============
+
+Library of mathematical functions operating with signed 64.64-bit fixed point
+numbers.
+
+Copyright (c) 2019, [ABDK Consulting](https://abdk.consulting/)
+
+All rights reserved.
+
+Signed 64.64 Fixed Point Numbers
+--------------------------------
+
+Signed 64.64-bit fixed point number is basically a simple fraction whose
+numerator is a signed 128-bit integer and denominator is 2^64.  As long as
+denominator is always the same, there is no need to store it, thus in Solidity
+signed 64.64-bit fixed point numbers are represented by int128 type holding only
+the numerator.
+
+Signed 64.64-bit fixed point numbers combine wide range and decent precision
+with good performance, and should be a good fit for most applications that need
+to deal with real numbers.
+
+Simple Arithmetic
+-----------------
+
+Here is the list of simple arithmetic functions provided by the library.
+
+    function add (int128 x, int128 y) internal pure returns (int128)
+
+Add one signed 64.64 fixed point number to another and return the result as a
+signed 64.64 fixed point number.
+
+    function sub (int128 x, int128 y) internal pure returns (int128)
+
+Subtract one signed 64.64 fixed point number from another and return the result
+as a signed 64.64 fixed point number.
+
+    function mul (int128 x, int128 y) internal pure returns (int128)
+
+Multiply one signed 64.64 fixed point number by another and return the result as
+a signed 64.64 fixed point number.
+
+    function muli (int128 x, int256 y) internal pure returns (int256)
+
+Multiply a signed 64.64 fixed point number by a signed 256-bit integer number
+and return the result as a signed 256-bit integer number.
+
+    function mulu (int128 x, uint256 y) internal pure returns (uint256)
+
+Multiply a signed 64.64 fixed point number by an unsigned 256-bit integer number
+and return the result as an unsigned 256-bit integer number.
+
+    function div (int128 x, int128 y) internal pure returns (int128)
+
+Divide one signed 64.64 fixed point number by another and return the result as
+a signed 64.64 fixed point number.
+
+    function divi (int256 x, int256 y) internal pure returns (int128)
+
+Divide one signed 256-bit integer number by another and return the result as a
+signed 64.64 fixed point number.
+
+    function divu (uint256 x, uint256 y) internal pure returns (int128)
+
+Divide one unsigned 256-bit integer number by another and return the result as
+a signed 64.64 fixed point number.
+
+    function neg (int128 x) internal pure returns (int128)
+
+Calculate the opposite for a signed 64.64 fixed point number, i.e. `-x`, and
+return the result as a signed 64.64 fixed point number.
+
+    function abs (int128 x) internal pure returns (int128)
+
+Calculate absolute value of a signed 64.64 fixed point number and return the
+result as a signed 64.64 fixed point number.
+
+    function inv (int128 x) internal pure returns (int128)
+
+Calculate the reciprocal for a signed 64.64 fixed point number, i.e. `1/x`, and
+return the result as a signed 64.64 fixed point number.
+
+Average Values
+--------------
+
+Here are average value functions.
+
+    function avg (int128 x, int128 y) internal pure returns (int128)
+
+Calculate the arithmetic average of two signed 64.64 fixed point numbers and
+return the result as a signed 64.64 fixed point number.
+
+    function gavg (int128 x, int128 y) internal pure returns (int128)
+
+Calculate the geometric average of two signed 64.64 fixed point numbers and
+return the result as a signed 64.64 fixed point number.
+
+Power and Root
+--------------
+
+Here are power and root functions.
+
+    function pow (int128 x, uint256 y) internal pure returns (int128)
+
+Raise a signed 64.64 fixed point number to a non-negative integer power and
+return the result as a signed 64.64 fixed point number.
+
+    function sqrt (int128 x) internal pure returns (int128)
+
+Calculate the square root of a signed 64.64 fixed point number and return the
+result as a signed 64.64 fixed point number.
+
+Exponentiation and Logarithm
+----------------------------
+
+Here are exponential and logarithm functions.
+
+    function log_2 (int128 x) internal pure returns (int128)
+
+Calculate the binary logarithm of a signed 64.64 fixed point number and return
+the result as a signed 64.64 fixed point number.
+
+    function ln (int128 x) internal pure returns (int128)
+
+Calculate the natural logarithm of a signed 64.64 fixed point number and return
+the result as a signed 64.64 fixed point number.
+
+    function exp_2 (int128 x) internal pure returns (int128)
+
+Raise 2 to the power of a signed 64.64 fixed point number, i.e. calculate
+`2^x`, and return the result as a signed 64.64 fixed point number.
+
+    function exp (int128 x) internal pure returns (int128)
+
+Exponentiate a signed 64.64 fixed point number, i.e. calculate `e^x`, and
+return the result as a signed 64.64 fixed point number.
+
+Conversions
+-----------
+
+Here are conversion functions.
+
+    function fromInt (int256 x) internal pure returns (int128)
+
+Convert a signed 256 bit integer number into a signed 64.64 bit fixed point
+number.
+
+    function toInt (int128 x) internal pure returns (int64)
+
+Convert a signed 64.64 bit fixed point number into a signed 64 bit integer
+number.
+
+    function fromUInt (uint256 x) internal pure returns (int128)
+
+Convert an unsigned 256 bit integer number into a signed 64.64 bit fixed point
+number.
+
+    function toUInt (int128 x) internal pure returns (uint64)
+
+Convert a signed 64.64 bit fixed point number into a unsigned 64 bit integer
+number.
+
+    function from128x128 (int256 x) internal pure returns (int128)
+
+Convert a signed 128.128 bit fixed point number into a signed 64.64 bit fixed
+point number.
+
+    function to128x128 (int128 x) internal pure returns (int256)
+
+Convert a signed 64.64 bit fixed point number into a signed 128.128 bit fixed
+point number.