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## Synopsis

Support for the full numeric tower.

## Interface

This extension provides support for large exact integers, exact rational and complex numbers.

The following standard procedures are redefined:

+     -      *     /
=     >      <     >=    <=
abs   max    min
eqv?  equal?
exp   expt   log   sin   cos   tan   atan   asin   acos   sqrt
quotient     modulo      remainder
exact?       inexact?
exact->inexact           inexact->exact
positive?    negative?   even?       odd?   zero?
number?      complex?    real?       rational?     integer?
gcd   lcm
truncate     ceiling     floor       round
number->string           string->number
numerator    denominator
rationalize
magnitude         angle
real-part         imag-part

The following standard procedures are provided:

make-rectangular  make-polar

The following non-standard extended procedures are redefined:

add1         sub1
signum
bitwise-and  bitwise-ior  bitwise-xor  bitwise-not  arithmetic-shift
randomize    random       finite?

Additionally the following procedures are available:

[procedure] (infinite? Z)

Is the number Z +inf.0 or -inf.0, or a complex number containing either infinity as its real or imaginary part?

Note: This is not the inverse of finite?, since it produces #f on nan.0 (or complex numbers containing nan.0), while finite? also returns #f in that case.

[procedure] (nan? Z)

Is the object Z a real representing nan.0?

[procedure] (exact-integer-sqrt K)

Returns two values s and r, where s^2 + r = K and K < (s+1)^2. In other words, s is the closest square root we can find that's equal to or smaller than K, and r is the rest if K isn't a neat square of two numbers.

[procedure] (exact-integer-nth-root K N)

Like exact-integer-sqrt, but with any base value. Calculates \sqrt[N]{K}, the Nth root of K and returns two values s and r where s^N + r = K and K < (s+1)^N.

[procedure] (exact-integer? X)

Is X an exact integer?

[procedure] (quotient&remainder A B)

Return the quotient and the remainder of A divided by B.

This is especially useful for bignums, since both numbers are derived simultaneously. This saves performing the division algorithm twice.

[procedure] (quotient&modulo A B)

Like quotient&remainder, except return the modulo instead of remainder.

[procedure] (conj Z)

Returns the conjugate of the complex number Z.

[procedure] (bignum? X)

Is X an extended-precision integer?

[procedure] (ratnum? X)

Is X a ratio?

[procedure] (cplxnum? X)

Is X a complex?

[procedure] (rectnum? X)

Is X an exact-complex? (Treats an integer-floatingpoint as "exact".)

[procedure] (compnum? X)

Is X an inexact-complex?

[procedure] (cflonum? X)

Is X a floatingpoint-complex or a floatingpoint?

[procedure] (cintnum? X)

Is X an integer-complex or an integer?

[procedure] (integer-length X)

From SRFI-60; get the number of bits required to represent the integer X in 2s complement notation.

## Example

(define (fac n)
(if (zero? n)
1
(* n (fac (- n 1))) ) ) ) )

(fac 100)
; => 9332621544394415268169923885626670049071596826438162146859296389
5217599993229915608941463976156518286253697920827223758251185210
916864000000000000000000000000

## Compiled code

Starting with version 2.8, extended number literals can be used in compiled code. To make this work, compile your code with:

 csc -X numbers-syntax foo.scm

IMPORTANT: This only works when the code will be run on exactly the same platform as the Scheme compiler ran on. Cross-compilation and compiling to C and compiling that on the target platform is not supported. (You will get an error message when you try to do it anyway)

## Bugs and limitations

• Extended number literals can not be used portably in compiled code (use string->number for maximum portability).
• For best results, use Chicken 4.7.4 or later. Extended numbers syntax isn't reliably supported by the reader of earlier Chicken versions (here as well, use string->number for maximum portability).

### Author

felix

The code for complex arithmetic was mostly taken from Thomas Chust's complex egg.

The CHICKEN Team

### Version history

trunk
Correct handling of exactness prefix (requires recent Chicken to work, at least 4.7.4). Division by inexact zero is no longer considered an error, but returns NaN or +Inf/-Inf. inexact->exact raises an error on NaN instead of returning 0 (reported by Felix Winkelmann). Fix log so it doesn't erroneously fail on complex numbers. Generalize asin and acos so they can return complex numbers. Fix <= and >= so they work when given a nan argument in rest position. Various fixes for code that expected inexact numbers so it calls exact->inexact before. Add types database for scrutinizer (Only in Chicken 4.7.4 and later).
2.7
Fix several bugs (expt, integer?, rational?, <=, >=, eqv?, -) found by importing the number tests from Gauche and writing an extensive test for number syntax edge cases. Complete rewrite of number parser (it should fully conform to the R7RS superset of R5RS number syntax now). Dropped dependency on the regex egg. Improved precision of exact->inexact so it can handle more extreme numbers. Provide the nan?, finite? and infinite? predicates and integer-length procedure.
2.6.1
Fix string->number so it raises an exception instead of crashing when passed a negative, zero or too large base (reported by Peter Hendrickson). Update test scripts so they exit with nonzero status in case of failed tests (reported by Mario Goulart)
2.6
Fix expt and log so they work for numbers which produce complex results.
2.5
Fix expt so it doesn't use flonums and loses precision in case of bignums
2.4
Added regex to dependency list
2.3
Added regex requirement to make it work in Chicken 4.6.2+
2.2
Fixed ratnum normalization in case of negative divisors
2.1
Changed quotient, remainder and quotient&remainder semantics so they accept fractional flonums to match Chicken's implementation of quotient and remainder. This also affects modulo and quotient&modulo.
2.0
Removed dependency on GMP, replacing it by the Scheme48 numbers code. Improved performance by integrating with GC. Added benchmarks and more tests. Converted testsuite to use the test egg instead of testeez. [by Peter Bex]
1.82
removed dependency on easyffi (Thanks to Peter Bex)
1.81
fixed dependencies
1.809
Fix for (expt <number> <ratio>), would fail [Bug noted by John Cowan]
1.808
Fix for 1.0+1i not treated as a rectnum or compnum [Bug noted by John Cowan] Added 'cintnum?' & 'cplxnum?' [kon lovett]
1.806
Fix for equal? when given keywords as args [Bug noted by papasi] [elf]
1.804
Using more accurate exact->inexact conversion for rationals [Alex Shinn]
1.803
fixed bug in fixnum bitwise operations [Thanks to Jeremy Sydik]
1.802
not quite complete support for static linking
1.801
Fixed a bug in angle [zb, mario]
1.8
Single-argument + and * didn't check argument type [Thanks to Stephen Gilardi]
1.701
Bugfix for compiler macros, undefined symbol [Kon Lovett]
1.7
Added compiler macros for bitwise-... operations
1.6
Compiler macros now work without -X numbers
1.5
Added support for speculative inlining of fixnum ops
1.4
Fixed bug in ratnum/bignum division [Thanks to Ivan Shmakov]
1.3
Fixing round for rational numbers [Alex Shinn]
1.2
Added bignum? ratnum? cflonum? rectnum? compnum? [Kon Lovett]
1.1
Uses easyffi properly [reported by John Cowan]
1.0
Fix for fixnum/rational comparison [by Zbigniew]
0.999
rational? always returned false for flonums
0.998
Added some missing definitions to numbers.scm [found by Kon Lovett]
0.997
Fixed memory leak in some bitwise operations [found by Dan Muresan]
0.996
Fixed a bug in expt [Daishi Kato]
0.995
round rounds to even (as specified by R5RS) [Thanks to Benedikt Rosenau]
0.994
Reimplemented = in C
0.993
Fixed equal? which didn't handle extended numbers nested in other data
0.992
Slight performance tuning for expt [Again by Daishi]
0.991
Fixed bug in fixnum/bignum subtraction [Thanks to Daishi Kato and Alex Shinn]
0.99
Speed improvements for expt by Daishi Kato
0.98
Added random and randomize [Suggested by Daishi Kato]
0.97
bignum/fixnum subtraction used wrong argument order [Thanks to Kon Lovett]
0.96
Several bug fixes by Alex Shinn; signum is exactness preserving
0.95
Alex Shinn contributed a working version of expt
0.94
Yet another bug-fix by Michal
0.93
Several bug-fixes by Michal Janeczek
0.92
exactness handling of expt is slightly better but still not perfect.
0.91
Fixed bug in integer?
0.9
(beta)

Copyright (c) 2008-2012 The CHICKEN Team
Copyright (c) 2000-2007, Felix L. Winkelmann
Copyright (c) 1993-2008 Richard Kelsey and Jonathan Rees
Copyright 1986,1987,1988,1989,1990,1991 Massachusetts Institute of Technology
Copyright 1992,1993,1994,2004 Massachusetts Institute of Technology

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