SafeDecimalMath¶
Description¶
This is a library contract that provides the ability to manipulate fractional numbers, performing safe arithmetic with unsigned fixedpoint decimals.
The decimals this library provides can operate at either of two different precision levels. Standard precision operations act on numbers with 18 decimal places. High precision numbers possess 27 decimal places, and have their own corresponding set of functions.^{1}
Several functions are included for converting between precision levels, and operations which round to the nearest increment to remove truncation bias. In SafeDecimalMath, a halfincrement rounds up.
SafeDecimalMath uses OpenZeppelin's SafeMath library for most of its basic arithmetic operations in order to protect from arithmetic overflows and zero divisions.
In Synthetix, the standard precision fixed point numbers are used to deal with most fractional quantities, such as token balances and prices. The highprecision numbers are mainly used for dealing with the debt ledger, which is constructed as an extended product of many fractional numbers very close to 1. As this is a financiallysensitive component of the system, representational precision matters in order to minimise errors resulting from rounding or truncation.
FixedPoint Mechanics¶
Representation¶
For a precision of d deimal places, this fixed point library chooses a large integer \dot{u} = 10^d to represent the number 1 (e.g. UNIT
= 10^{18}) and all operations at this precision level happen relative to \dot{u}. That is, the fixed point representation of a number q is defined to be the integer \dot{q}:
For example, at 27 decimal places, \dot{25} is equivalent to 25 \times 10^{27}. We will use square brackets to capture the fixed point representation of composite expressions.
Note that this is only valid if \dot{q} is an integer, so nothing is representable which has a positive value in the decimal places smaller than \frac{1}{\dot{u}} (i.e. the integer 1).
Operations¶
We define the fixed point operations \dot{+}, \dot{}, \dot{\times}, \dot{/}, corresponding to the ordinary arithmetic operations +, , \times, /, where / corresponds to integer division. These are implemented by SafeMath
and protect from overflow.
Additive Operations¶
We define our additive fixed point operators to be the same as the standard ones:
Definition: Fixed Point Addition and Subtraction
This is because:
Multiplicative Operations¶
The multiplicative operations are defined as follows:
Definition: Fixed Point Multiplication and Division
Some care has to be taken for multiplication and division. We desire, for example, \dot{p} \ \dot{\times} \ \dot{q} = \dot{[p \times q]}. However, if the standard operations are performed naively, the following results are obtained:
So multiplication produces an extra unwanted unit factor, and division divides one out; the fixed point operations need to account for this. Note that to ensure minimum precision loss, \dot{u} is divided out last in the case of multiplication and multiplied in first in the case of division.
Rounding¶
Note that multiplication and division of fixed point numbers may involve some loss of precision in the lowest digit. Such inaccuracy can accumulate over many operations
Synthetix provides versions of \dot{\times} and \dot{/} which perform the operation with one extra internal digit of precision, and then rounds up if the least significant digit is 5 or greater. Consequently, results exactly halfway between two increments are rounded up.
Change of Precision¶
The representation of a number q at two different fixed point precision levels \dot{q} = q \dot{u} and \ddot{q} = q \ddot{u} is straightforward if \dot{u} and \ddot{u} divide evenly. If this is the case, and \ddot{u} is the higher precision unit, then \ddot{q} / \dot{q} = \ddot{u} / \dot{u}. So converting between the high and low precision only involves multiplying or dividing by a factor of \ddot{u} / \dot{u}. Keep in mind that converting from a high precision to a low precision number involves some loss of information, and this operation is performed with rounding.
Source: SafeDecimalMath.sol
Architecture¶
Inheritance Graph¶
Libraries¶
 SafeMath for
uint
Variables¶
decimals
¶
The number of decimals (18) in the standard precision fixed point representation.
Type: uint8 public constant
Value: 18
highPrecisionDecimals
¶
The number of decimals (27) in the high precision fixed point representation.
Type: uint8 public constant
Value: 27
UNIT
¶
The standard precision number (10^{18}) that represents 1.0.
Type: uint public constant
Value: 1e18
PRECISE_UNIT
¶
The high precision number (10^{27}) that represents 1.0.
Type: uint public constant
Value: 1e27
UNIT_TO_HIGH_PRECISION_CONVERSION_FACTOR
¶
The factor (10^9) to convert between precision levels. Equivalent to PRECISE_UNIT / UNIT
.
Type: uint private constant
Value: 1e9
Functions¶
unit
¶
Pure alias to UNIT
.
Details
Signature
unit() external pure returns (uint)
preciseUnit
¶
Pure alias to PRECISE_UNIT
.
Details
Signature
preciseUnit() external pure returns (uint)
multiplyDecimal
¶
Returns the product of two standard precision fixed point numbers, handling precision loss by truncation.
Details
Signature
multiplyDecimal(uint x, uint y) internal pure returns (uint)
_multiplyDecimalRound
¶
Returns the product of two fixed point numbers, handling precision loss by rounding. This function is private, and takes the fixedpoint precision as a parameter, only being used to implement multiplyDecimalRound
and multiplyDecimalRoundPrecise
.
Details
Signature
_multiplyDecimalRound(uint x, uint y, uint precisionUnit) private pure returns (uint)
multiplyDecimalRoundPrecise
¶
Returns the product of two high precision fixed point numbers, handling precision loss by rounding.
Equivalent to _multiplyDecimalRound(x, y, PRECISE_UNIT)
.
Details
Signature
multiplyDecimalRoundPrecise(uint x, uint y) internal pure returns (uint)
multiplyDecimalRound
¶
Returns the product of two standard precision fixed point numbers, handling precision loss by rounding.
Equivalent to _multiplyDecimalRound(x, y, UNIT)
.
Details
Signature
multiplyDecimalRound(uint x, uint y) internal pure returns (uint)
divideDecimal
¶
Returns the quotient of two standard precision fixed point numbers, handling precision loss by truncation.
Details
Signature
divideDecimal(uint x, uint y) internal pure returns (uint)
_divideDecimalRound
¶
Returns the quotient of two fixed point numbers, handling precision loss by rounding. This function is private, and takes the fixedpoint precision as a parameter, only being used to implement divideDecimalRound
and divideDecimalRoundPrecise
.
Details
Signature
_divideDecimalRound(uint x, uint y, uint precisionUnit) private pure returns (uint)
divideDecimalRound
¶
Returns the quotient of two standard precision fixed point numbers, handling precision loss by rounding.
Equivalent to _divideDecimalRound(x, y, UNIT)
.
Details
Signature
divideDecimalRound(uint x, uint y) internal pure returns (uint)
divideDecimalRoundPrecise
¶
Returns the quotient of two high precision fixed point numbers, handling precision loss by rounding.
Equivalent to _divideDecimalRound(x, y, PRECISE_UNIT)
.
Details
Signature
divideDecimalRoundPrecise(uint x, uint y) internal pure returns (uint)
decimalToPreciseDecimal
¶
Converts from standard precision to high precision numbers. This is just multiplication by 10^9.
Details
Signature
decimalToPreciseDecimal(uint i) internal pure returns (uint)
preciseDecimalToDecimal
¶
Converts from high precision to standard precision numbers. This is division by 10^9, where precision loss is handled by rounding.
Details
Signature
preciseDecimalToDecimal(uint i) internal pure returns (uint)

SafeDecimalMath provides two different precision levels because the Ethereum virtual machine encodes integers in 256 bits. Given a finite integer size, an increase in precision implies a decrease in the maximum representable number, since more bits are dedicated to representing the fractional part, and fewer to the integer part. ↩