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Rounding modes

Fastly VCL provides access to various rounding modes by way of independent functions for rounding values. These functions have explicit rounding modes. There is no stateful interface to set a "current" rounding mode.

Fastly VCL does not provide interfaces to round values to a given number of significant figures, to a given multiple, or to a given power.

Tie-breaking when rounding to nearest

The roundoff errors introduced by rounding values to their nearest integers are symmetric, except for treatment of the exact midpoint between adjacent integers.

That is, for every value that gets rounded up (such as 3.77 rounding up to the nearest integer 4.0), there is a corresponding value (3.23) which is rounded down by the same amount. This can be seen visually:

tie-breaking

Rounding to the nearest integer requires a tie-breaking rule for when the fractional part of a value is exactly 0.5. There are several ways to break these ties, enumerated in the "to nearest" rounding modes below.

Overview

Example values:

Inputceilfloortruncroundroundevenroundhalfuproundhalfdown
-1.8-1.0-2.0-1.0-2.0-2.0-2.0-2.0
-1.5-1.0-2.0-1.0-2.0-2.0-1.0-2.0
-1.2-1.0-2.0-1.0-1.0-1.0-1.0-1.0
-0.5-0.0-1.0-0.0-1.0-0.0-0.0-1.0
0.51.00.00.01.00.01.00.0
1.22.01.01.01.01.01.01.0
1.52.01.01.02.02.02.01.0
1.82.01.01.02.02.02.02.0

A visual representation of the same:

rounding-modes

"Direct" rounding modes

All of the following modes always round towards a specific direction.

  • Round up — math.ceil
    Also known as ceiling, round towards positive infinity IEEE 754 roundTowardPositive

    Non-integer values are rounded up towards +∞. Negative results thus round toward zero.

  • Round down — math.floor
    Also known as floor, round towards negative infinity IEEE 754 roundTowardNegative

    Non-integer values are rounded down towards -∞. Negative results thus round away from zero.

  • Round towards zero — math.trunc
    Also known as truncation, round away from infinity IEEE 754 roundTowardZero

    Rounding is performed by removing the fractional part of a number, leaving the integral part unchanged.

    Note the FLOAT to INTEGER type conversion in Fastly VCL is not by truncation (as it is in many comparable languages). See discussion under ties away from zero.

  • Round away from zero
    Also known as round towards infinity

    Positive non-integer values are rounded up towards positive infinity. Negative non-integer values are rounded down towards negative infinity.

    Not provided in Fastly VCL.

“To nearest” rounding modes

All of the following modes round non-tie values to their nearest integer. These modes differ only in their treatment of ties.

  • Round to nearest, ties away from zero — math.round
    Also known as commercial rounding IEEE 754 roundTiesToAway

    For positive values, ties are rounded up towards positive infinity. For negative values, ties are rounded down towards negative infinity.

    This is symmetric behaviour, avoiding bias to either positive or negative values. However, this mode does introduce bias away from zero.

    This rounding mode is used for implicit FLOAT to INTEGER type conversions in VCL. These behave as if by a call to math.round.

  • Round to nearest, ties to even — math.roundeven
    Also known as half to even, convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, and bankers' rounding IEEE 754 roundTiesToEven

    Of the two nearest integer values, ties are rounded either up or down to whichever value is even.

    This rounding mode increases the probability of even numbers relative to odd numbers, but avoids bias to either positive or negative values, and also avoids bias towards or away from zero. The cumulative error is minimized when summing rounded values, especially when the values are predominantly positive or predominantly negative.

  • Round to nearest, ties towards positive infinity — math.roundhalfup
    Also known as half up

    This is asymmetric behavior, where ties for negative values are rounded towards zero, and ties for positive values are rounded away from zero.

WARNING: Some languages use the term half up to mean symmetric behaviour. For rounding functions in these languages, "up" is a value of larger absolute magnitude. That is, negative ties will be rounded away from zero, which differs from the behavior in VCL. Take care when porting code using this rounding mode to VCL.

  • Round to nearest, ties towards negative infinity — math.roundhalfdown
    Also known as half down

    This is asymmetric behavior, where ties for negative values are rounded away from zero, and ties for positive values are rounded towards zero.

WARNING: Some languages use the term half down to mean symmetric behavior. For rounding functions in these languages, "down" is a value of smaller absolute magnitude. That is, negative ties will be rounded towards zero, which differs from the behavior in VCL. Take care when porting code using this rounding mode to VCL.

  • Round to nearest with other tie-breaking schemes
    There are several other less common arrangements for tie-breaking. These include ties to odd (in a similar manner as ties to even), random tie-breaking, and stochastic tie-breaking.

    These schemes are not provided in Fastly VCL.

Floating point numbers have have more computational nuances than are described by the cursory discussion of biases here. For more details, see What every computer scientist should know about floating-point arithmetic

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