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Numbers

This egg provides support for the full numeric tower. It adds support for:

All of these are supported by exporting different versions of the standard numerical procedures. Be sure to import the numbers module after the scheme module, or to hide the standard numerical procedures from scheme. These number types are also supported as read-time literals, but only after loading the extension and only in the interpreter (but see below for limited compile-time support).

This egg also adds a handful of extra procedures. See below.

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 log procedure is extended to allow a second argument specifying the logarithm base, as per R7RS.

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?      bit-set?

Additionally the following procedures are available:

infinite? Zprocedure

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

From R6RS and R7RS.

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.

nan? Zprocedure

Is the object Z a real representing nan.0?

From R6RS and R7RS.

exact Zprocedure
inexact Zprocedure

Convert Z to an exact or an inexact number, respectively. These are just the R6RS and R7RS names for inexact->exact and exact->inexact.

exact-integer-sqrt Kprocedure

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.

This procedure is compatible with the R7RS specification.

exact-integer-nth-root K Nprocedure

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.

exact-integer? Xprocedure

Is X an exact integer?

quotient&remainder A Bprocedure

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.

This name is deprecated in favor of its more "regular" alias from R7RS, truncate/.

quotient&modulo A Bprocedure

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

This procedure has been deprecated.

floor/ A Bprocedure
floor-quotient A Bprocedure
floor-remainder A Bprocedure
truncate/ A Bprocedure
truncate-quotient A Bprocedure
truncate-remainder A Bprocedure

These procedures are from R7RS.

The floor/ and truncate/ versions return two values: the quotient and remainder obtained after dividing A by B. The -quotient and -remainder procedures simply return only the corresponding value of the "full" computation.

The difference is in how the values are derived for negative values.

If both values are desired, the two-value versions are recommended, as they perform the computation only once.

All the procedures compute a quotient q and remainder r such that A = nq + r. The remainder is determined by the choice for q, through the relation r = A - nq. The floor procedures compute q = floor(A/B) and the truncate procedures compute q = truncate(A/B).

See Riastradh's proposal for Division operators in Scheme for a full explanation of the intricacies of these procedures.

conj Zprocedure

Returns the conjugate of the complex number Z.

bignum? Xprocedure

Is X an extended-precision integer?

ratnum? Xprocedure

Is X a ratio?

cplxnum? Xprocedure

Is X a complex?

rectnum? Xprocedure

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

compnum? Xprocedure

Is X an inexact-complex?

cflonum? Xprocedure

Is X a floatingpoint-complex or a floatingpoint?

cintnum? Xprocedure

Is X an integer-complex or an integer?

integer-length Xprocedure

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: Extended number literals only work 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

About this egg

Author

felix

The code for complex arithmetic was mostly taken from Thomas Chust's complex egg for CHICKEN 3.

Maintainer

The CHICKEN Team

Version history

4.6.3
Fix a bug introduced in 4.6.2 related to printing negative floating-point numbers.
4.6.2
Fix an edge case in printing floating-point numbers resulting in 0.0 for numbers that will be normalized to 2^64.
4.6.1
Fix a few warnings during compilation due to invalid exclude directives in module definition (thanks to Evan Hanson).
4.6
Proper fix for Burnikel-Ziegler performance problem from release 4.3, thereby fixing another pathological case.
4.5
Fix "overlap" handling in number->string for power of two radices which have a bit length that's not a multiple of the word size.
4.4
Add support for "argvector" CHICKEN, while maintaining support for pre-argvector versions.
4.3
Fix pathological performance problem in Burnikel-Ziegler division routines triggered by "pidigits" benchmark from the Great Languages Shootout Game (thanks to "Balkenbrij" on Reddit). Fix a double C_fix() call in bitwise-and that would sometimes cause invalid result values due to 1 limb of uninitialized memory. Fix a similar bug in comparisons of ratnums with other numbers. Fix error reporting when passed non-numeric objects to elementary arithmetic operators. Make bignum to floating-point more stable across platforms by disabling extended precision FP registers (on Linux/i386).
4.2
Fix an error in Burnikel-Ziegler division that caused wrong results in some edge cases. Fix compilation error in pre-4.8.0 CHICKENs due to use of missing C macro.
4.1
Fix a few errors in specialization rules and a strange use of "eval" in the unit tests.
4.0
Convert internal digit representation to use full words. Improve performance by accessing halfdigits directly. Improve performance of number<->string conversions for radixes that are a power of two by directly operating on digits, avoiding in-place shifts. Improve performance of number->string by using a divide and conquer algorithm. Implement Burnikel&Ziegler recursive division which is O(r*s^{log(3)-1} + r*log(s)) instead of O(n^2) (this is mostly useful for very large numbers due to high constant overhead). Make comparisons inlineable and implement vararg comparison procedures in C to avoid consing up a rest list. Implement Karatsuba square root, which is more efficient than Newton's algorithm. Implement bit-set?. Rewrite the last remaining s48 code (very little was left) to ensure clean license/copyright.
3.1
Improve performance of string<->number conversions by chunking bignum digits off instead of using a binary digit at a time. Use a faster and shorter division algorithm from Hacker's Delight. Implement Lehmer's hybrid Euclidian GCD algorithm which is O(n^2/log n) instead of O(n^2) like Euclid and Stein's algorithms. Implement Karatsuba multiplication which is O(n^log2(3)) instead of O(n^2), but is only effective for larger bignums. Make a few more things inlineable. Fix stupid mistake which caused exact operations involving rational numbers to be needlessly slow. Improve performance for string<->number conversion using radix powers of two, and for multiplication of bignums with small powers of two.
3.0.1
Fix quotient&remainder and derived procedures for 2 bignum arguments (it returned only the remainder). Fix crash bug in flonum comparison.
3.0
Improve performance of exact-integer-sqrt and exact-integer-nth-root by using a better initial guess. Improve read/write performance. Major refactoring to make things a lot more consistent with the core system. Added some more specializations. Change integer-length, bitwise-and, bitwise-ior, bitwise-ior, bitwise-not, arithmetic-shift and integer-length to only allow exact integers. Handling of rationals should be better, as well as a few other division procedures.
2.10.1
Fix a few inexact number tests to use nonzero epsilons.
2.10
Fix error reporting of >=, <=, nan?, finite? and infinite? [thanks to Felix Winkelmann]. Fix types DB entry for inexact->exact (found by adding insanely paranoid checks to CHICKEN itself)
2.9
Fix a few compiler warnings. R7RS support: Add support for a 2nd "base" argument to the log procedure, export exact and inexact aliases for exact->inexact and inexact->exact. Add floor/, floor-quotient, floor-remainder, truncate/, truncate-quotient and truncate-remainder.
2.8.2
Add CL-compatible support for complex numbers to signum (#953), and support numerator and denominator on flonums (as per RnRS) (#1016) [Thanks to John Cowan]
2.8.1
Remove long-deprecated and undocumented numbers:-prefixed variants of +, -, etc. Fix compilation on 4.8.1, which failed due to invalid +inf/-inf literals. Remove (declare (disable-interrupts)), which should make numbers play better with threaded code.
2.8
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)

License

Copyright (c) 2008-2017 The CHICKEN Team
Copyright (c) 2000-2007, Felix L. Winkelmann
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
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   notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
   notice, this list of conditions and the following disclaimer in the
   documentation and/or other materials provided with the distribution.
3. The name of the authors may not be used to endorse or promote products
   derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY DIRECT, INDIRECT,
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NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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