The Input Format

Herbie's input format is designed for expressing mathematical functions, which Herbie can then search for accurate implementations of. It also allows specifying the distribution that Herbie draws inputs from when evaluating the accuracy of an expression.

General format

Herbie uses the FPCore format for its input expression, which looks like this:

(FPCore (inputs ...) properties ... expression)

Each input is a variable, like x, which can be used in the expression, whose accuracy Herbie will try to improve. Properties are described below.

The expression is written in prefix form, with every function call parenthesized, as in Lisp. For example, the formula for the hypotenuse of a triangle with legs a and b is

(FPCore (a b) (sqrt (+ (* a a) (* b b))))

We recommend the .fpcore file extension for Herbie input files.

Supported functions

Herbie supports all functions from math.h with floating-point-only inputs and outputs. The best supported functions, far from the full list, include:

+, -, *, /, fabs

The usual arithmetic functions
- is both negation and subtraction

sqrt, cbrt

Square and cube roots

pow, exp, log

Various exponentiations and logarithms

sin, cos, tan

The trigonometric functions

asin, acos, atan, atan2

The inverse trigonometric functions

sinh, cosh, tanh

The hyperbolic functions

asinh, acosh, atanh

The inverse hyperbolic functions

fma, expm1, log1p, hypot

Specialized numeric functions

Herbie also supports the constants PI and E.

Herbie links against libm to ensure that every function has the same behavior in Herbie as in your code. However, on Windows platforms some functions are not available in the system libm. In these cases Herbie will use a fallback implementation and print a warning; turning off the the precision:fallback option disables those functions instead.

Conditionals

FPCore uses if for conditional expressions:

(if cond if-true if-false)

An if epxression evaluates the conditional cond and returns either if-true if it is true or if-false if it is not. Conditionals may use:

==, !=, <, >, <=, >=

The usual comparison operators

and, or, not

The usual logical operators

TRUE, FALSE

The two boolean values

Note that unlike the arithmetic operators, the comparison functions can take any number of arguments.

Intermediate variables

Intermediate variables can be defined using let:

(let ([variable value] ...) body)

In a let expression, all the values are evaluated first, and then are bound to their variables in the body. This means that the value of one variable can't refer to another variable in the same let block; nest let constructs if you want to do that.

Note that Herbie treats intermediate values only as a notational convenience, and inlines their values before improving the formula's accuracy. Using intermediate variables will not help Herbie improve a formula's accuracy or speed up its run-time.

Complex Numbers^β

Herbie includes experimental support for complex numbers; however, this support is currently limited to the basic arithmetic operations. Some of Herbie's internal mechanisms for improving expression accuracy also do not yet support complex-number expressions.

All input parameters are real numbers; complex numbers must be constructed with complex. The functions +, -, *, /, re, im, and conj are available on complex numbers. Note that complex and real operations use the same syntax; however, complex and real arithmetic cannot be mixed: (+ (complex 1 2) 1) is not valid. A type checker will report such errors.

Complex operations use the Racket implementation, so results may differ (slightly) for the complex library used in your language, especially for non-finite complex numbers. Unfortunately, complex number arithmetic is not as standardized as float-point arithmetic.

In the future, we hope to support complex-number arguments and fully support all complex-number operations.

Properties

Herbie also allows several FPCore properties specified on inputs for additional meta-data:

:name string

Herbie uses this name in its output

:pre test

Herbie samples only points that pass the test in the reals

Several additional properties can be found in the benchmark suite and are used for testing, but are not supported and can change without warning.

Herbie's output uses custom FPCore properties in its output to provide meta-data about the Herbie improvement process:

:herbie-status status

Describes whether Herbie successfully improved the accuracy of the input; status is one of success, timeout, error, or crash.

:herbie-time ms

The time, in milliseconds, used by Herbie to find a more accurate formula.

:herbie-bits-used bits

The precision used to find accurate outputs from the formula.

:herbie-error-input ([pts err] ...)

The computed average error of the input program, evaluated on pts points. Multiple entries correspond to multiple training or test sets.

:herbie-error-output ([pts2 err1] [pts2 err2])

The computed average error of the output program, like above.

Herbie's output also passes through any :name and :pre properties on its inputs.

Converting from Herbie 0.9

Herbie 0.9 used a different input format, which is not supported Herbie 1.0 and later. To simplify the transition, the infra/convert.rkt script converts from the old to the new format.

To use the conversion tool, run:

racket infra/convert.rkt file.rkt > file.fpcore