The floating-point-contractions Reference Manual

Table of Contents

The floating-point-contractions Reference Manual

This is the floating-point-contractions Reference Manual, generated automatically by Declt version 3.0 "Montgomery Scott" on Fri Jun 26 11:05:14 2020 GMT+0.

1 Introduction

This system provides some numerically stable, contracted versions of
common but unstable floating-point operations.

| Export   | (AKA)    | Computes        |
|----------+----------+-----------------|
| log1+    | log1p    | log (1 + x)     |
| log1-    | log1m    | log (1 - x)     |
| log1+/x  |          | (log (1 + x))/x |
| exp-1    | expm1    | exp(x) - 1      |
| exp-1/x  |          | (exp(x)-1)/x    |
| expt-1   | powm1    | (x^y)-1         |
| log1-exp | log1mexp | log(1 - exp(x)) |
| log1+exp | log1pexp | log(1 + exp(x)) |
| log2-exp | log2mexp | log(2 - exp(x)) |
| logexp-1 | logexpm1 | log(exp(a) -1)  |
| hypot    |          | sqrt(x^2 + y^2) |

2 Systems

The main system appears first, followed by any subsystem dependency.

2.1 floating-point-contractions

Author

Paul M. Rodriguez <pmr@ruricolist.com>

License

MIT

Description

Numerically stable contractions of floating-point operations.

Source

floating-point-contractions.asd (file)

Components

• package.lisp (file)
• floating-point-contractions.lisp (file)

3 Files

Files are sorted by type and then listed depth-first from the systems components trees.

3.1 Lisp

3.1.1 floating-point-contractions.asd

Location

floating-point-contractions.asd

Systems

floating-point-contractions (system)

3.1.2 floating-point-contractions/package.lisp

Parent

floating-point-contractions (system)

Location

package.lisp

Packages

floating-point-contractions

3.1.3 floating-point-contractions/floating-point-contractions.lisp

Dependency

package.lisp (file)

Parent

floating-point-contractions (system)

Location

floating-point-contractions.lisp

Exported Definitions

• exp-1 (function)
• exp-1/x (function)
• expt-1 (function)
• hypot (function)
• lb (function)
• lg (function)
• ln (function)
• log1+ (function)
• log1+/x (function)
• log1+exp (function)
• log1- (function)
• log1-exp (function)
• log2-exp (function)
• logexp-1 (function)

Internal Definitions

sq (function)

4 Packages

Packages are listed by definition order.

4.1 floating-point-contractions

Source

package.lisp (file)

Use List

common-lisp

Exported Definitions

• exp-1 (function)
• exp-1/x (function)
• expt-1 (function)
• hypot (function)
• lb (function)
• lg (function)
• ln (function)
• log1+ (function)
• log1+/x (function)
• log1+exp (function)
• log1- (function)
• log1-exp (function)
• log2-exp (function)
• logexp-1 (function)

Internal Definitions

sq (function)

5 Definitions

Definitions are sorted by export status, category, package, and then by lexicographic order.

5.1 Exported definitions

5.1.1 Functions

Function: exp-1 X

Compute (- (exp x) 1) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: exp-1/x X

Compute (/ (- (exp x) 1) x) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: expt-1 A Z

Compute (a^z)-1 stably even when A is close to 1 or Z is close to zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: hypot X Y

Compute the hypotenuse of X and Y without danger of floating-point overflow or underflow.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: lb N

Binary logarithm.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: lg N

Decimal logarithm.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: ln N

Natural logarithm.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log1+ X

Compute (log (+ 1 x)) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log1+/x X

Compute (/ (log (+ 1 x)) x) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log1+exp A

Accurately compute log(1+exp(x)) even when A is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log1- X

Compute (log (- 1 x)) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log1-exp A

Compute log(1-exp(x)) stably even when A is near zero.

This is sometimes known as the E_3, the third Einstein function.

See Mächler 2008 for notes on accurate calculation.

https://cran.r-project.org/web/packages/Rmpfr/vignettes/log1mexp-note.pdf

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: log2-exp X

Compute log(2-exp(x)) stably even when X is near zero.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Function: logexp-1 A

Compute log(exp(a)-1) stably even when A is small.

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

5.2 Internal definitions

5.2.1 Functions

Function: sq X

Package

floating-point-contractions

Source

floating-point-contractions.lisp (file)

Appendix A Indexes

A.1 Concepts

A.2 Functions

A.3 Variables

A.4 Data types