Dappsys

Safe, simple, flexible building-blocks for smart-contract systems.

Auth: ds-auth ds-guard ds-roles
Token: ds-token ds-vault
Oracle: ds-cache ds-value
Util: ds-exec ds-math ds-note ds-proxy ds-stop ds-thing
Gov: ds-multisig

DSMath

Safe Arithmetic

DS-Math provides arithmetic functions for the common numerical primitive types of Solidity. You can safely add, subtract, multiply, and divide uint numbers without fear of integer overflow. You can also find the minimum and maximum of two numbers.

Additionally, this package provides arithmetic functions for new two higher level numerical concepts called wad (18 decimals) and ray (27 decimals). These are used to represent fixed-point decimal numbers.

A wad is a decimal number with 18 digits of precision and a ray is a decimal number with 27 digits of precision. These functions are necessary to account for the difference between how integer arithmetic behaves normally, and how decimal arithmetic should actually work. A brief example using wmul, which returns the product of a wad and another number:

1.1 * 2.2 == 2.42

//Regular integer arithmetic adds orders of magnitude:

110 * 220 == 24200

// Wad arithmetic does not add orders of magnitude:

wmul(1.1 ether, 2.2 ether) == 2.42 ether

Naming Convention

The standard functions are the uint set, so their function names are not prefixed: add, sub, mul, min, and max. There is no div function, as divide-by-zero checking is built into the Solidity compiler.

The int functions have an i prefix: imin, and imax.

Wad functions have a w prefix: wmul, wdiv.

Ray functions have a r prefix: rmul, rdiv, rpow.

API Reference

`add`

Return x + y or an exception in case of uint overflow.

`sub`

Return x - y or an exception in case of uint overflow.

`mul`

Return x * y or an exception in case of uint overflow.

`min`

Return the smaller number of x and y.

`max`

Return the larger number of x and y.

`imin`

Return the smaller number of x and y.

`imax`

Return the larger number of x and y.

`wmul`

Multiply two Wads and return a new Wad with the correct level of precision. A Wad is a decimal number with 18 digits of precision that is being represented as an integer.

`wdiv`

Divide two Wads and return a new Wad with the correct level of precision. A Wad is a decimal number with 18 digits of precision that is being represented as an integer.

`rmul`

Multiply two Rays and return a new Ray with the correct level of precision. A Ray is a decimal number with 27 digits of precision that is being represented as an integer.

`rdiv`

Divide two Rays and return a new Ray with the correct level of precision. A Ray is a decimal number with 27 digits of precision that is being represented as an integer.

`rpow`

Raise a Ray to the n^th power and return a new Ray with the correct level of precision. A Ray is a decimal number with 27 digits of precision that is being represented as an integer.

  /// math.sol -- mixin for inline numerical wizardry

// This program is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.

// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.

// You should have received a copy of the GNU General Public License
// along with this program.  If not, see <http://www.gnu.org/licenses/>.

pragma solidity ^0.4.13;

contract DSMath {
    function add(uint x, uint y) internal pure returns (uint z) {
        require((z = x + y) >= x);
    }
    function sub(uint x, uint y) internal pure returns (uint z) {
        require((z = x - y) <= x);
    }
    function mul(uint x, uint y) internal pure returns (uint z) {
        require(y == 0 || (z = x * y) / y == x);
    }

    function min(uint x, uint y) internal pure returns (uint z) {
        return x <= y ? x : y;
    }
    function max(uint x, uint y) internal pure returns (uint z) {
        return x >= y ? x : y;
    }
    function imin(int x, int y) internal pure returns (int z) {
        return x <= y ? x : y;
    }
    function imax(int x, int y) internal pure returns (int z) {
        return x >= y ? x : y;
    }

    uint constant WAD = 10 ** 18;
    uint constant RAY = 10 ** 27;

    function wmul(uint x, uint y) internal pure returns (uint z) {
        z = add(mul(x, y), WAD / 2) / WAD;
    }
    function rmul(uint x, uint y) internal pure returns (uint z) {
        z = add(mul(x, y), RAY / 2) / RAY;
    }
    function wdiv(uint x, uint y) internal pure returns (uint z) {
        z = add(mul(x, WAD), y / 2) / y;
    }
    function rdiv(uint x, uint y) internal pure returns (uint z) {
        z = add(mul(x, RAY), y / 2) / y;
    }

    // This famous algorithm is called "exponentiation by squaring"
    // and calculates x^n with x as fixed-point and n as regular unsigned.
    //
    // It's O(log n), instead of O(n) for naive repeated multiplication.
    //
    // These facts are why it works:
    //
    //  If n is even, then x^n = (x^2)^(n/2).
    //  If n is odd,  then x^n = x * x^(n-1),
    //   and applying the equation for even x gives
    //    x^n = x * (x^2)^((n-1) / 2).
    //
    //  Also, EVM division is flooring and
    //    floor[(n-1) / 2] = floor[n / 2].
    //
    function rpow(uint x, uint n) internal pure returns (uint z) {
        z = n % 2 != 0 ? x : RAY;

        for (n /= 2; n != 0; n /= 2) {
            x = rmul(x, x);

            if (n % 2 != 0) {
                z = rmul(z, x);
            }
        }
    }
}