1 ;;;; This file contains the definitions of float specific number
2 ;;;; support (other than irrational stuff, which is in irrat.) There is
3 ;;;; code in here that assumes there are only two float formats: IEEE
4 ;;;; single and double. (Long-float support has been added, but bugs
5 ;;;; may still remain due to old code which assumes this dichotomy.)
7 ;;;; This software is part of the SBCL system. See the README file for
10 ;;;; This software is derived from the CMU CL system, which was
11 ;;;; written at Carnegie Mellon University and released into the
12 ;;;; public domain. The software is in the public domain and is
13 ;;;; provided with absolutely no warranty. See the COPYING and CREDITS
14 ;;;; files for more information.
16 (in-package "SB!KERNEL")
20 (eval-when (:compile-toplevel :load-toplevel :execute)
22 ;;; These functions let us create floats from bits with the
23 ;;; significand uniformly represented as an integer. This is less
24 ;;; efficient for double floats, but is more convenient when making
25 ;;; special values, etc.
26 (defun single-from-bits (sign exp sig)
27 (declare (type bit sign) (type (unsigned-byte 24) sig)
28 (type (unsigned-byte 8) exp))
30 (dpb exp sb!vm:single-float-exponent-byte
31 (dpb sig sb!vm:single-float-significand-byte
32 (if (zerop sign) 0 -1)))))
33 (defun double-from-bits (sign exp sig)
34 (declare (type bit sign) (type (unsigned-byte 53) sig)
35 (type (unsigned-byte 11) exp))
36 (make-double-float (dpb exp sb!vm:double-float-exponent-byte
37 (dpb (ash sig -32) sb!vm:double-float-significand-byte
38 (if (zerop sign) 0 -1)))
39 (ldb (byte 32 0) sig)))
40 #!+(and long-float x86)
41 (defun long-from-bits (sign exp sig)
42 (declare (type bit sign) (type (unsigned-byte 64) sig)
43 (type (unsigned-byte 15) exp))
44 (make-long-float (logior (ash sign 15) exp)
45 (ldb (byte 32 32) sig)
46 (ldb (byte 32 0) sig)))
52 (defconstant least-positive-single-float (single-from-bits 0 0 1))
53 (defconstant least-positive-short-float least-positive-single-float)
54 (defconstant least-negative-single-float (single-from-bits 1 0 1))
55 (defconstant least-negative-short-float least-negative-single-float)
56 (defconstant least-positive-double-float (double-from-bits 0 0 1))
58 (defconstant least-positive-long-float least-positive-double-float)
59 #!+(and long-float x86)
60 (defconstant least-positive-long-float (long-from-bits 0 0 1))
61 (defconstant least-negative-double-float (double-from-bits 1 0 1))
63 (defconstant least-negative-long-float least-negative-double-float)
64 #!+(and long-float x86)
65 (defconstant least-negative-long-float (long-from-bits 1 0 1))
67 (defconstant least-positive-normalized-single-float
68 (single-from-bits 0 sb!vm:single-float-normal-exponent-min 0))
69 (defconstant least-positive-normalized-short-float
70 least-positive-normalized-single-float)
71 (defconstant least-negative-normalized-single-float
72 (single-from-bits 1 sb!vm:single-float-normal-exponent-min 0))
73 (defconstant least-negative-normalized-short-float
74 least-negative-normalized-single-float)
75 (defconstant least-positive-normalized-double-float
76 (double-from-bits 0 sb!vm:double-float-normal-exponent-min 0))
78 (defconstant least-positive-normalized-long-float
79 least-positive-normalized-double-float)
80 #!+(and long-float x86)
81 (defconstant least-positive-normalized-long-float
82 (long-from-bits 0 sb!vm:long-float-normal-exponent-min
83 (ash sb!vm:long-float-hidden-bit 32)))
84 (defconstant least-negative-normalized-double-float
85 (double-from-bits 1 sb!vm:double-float-normal-exponent-min 0))
87 (defconstant least-negative-normalized-long-float
88 least-negative-normalized-double-float)
89 #!+(and long-float x86)
90 (defconstant least-negative-normalized-long-float
91 (long-from-bits 1 sb!vm:long-float-normal-exponent-min
92 (ash sb!vm:long-float-hidden-bit 32)))
94 (defconstant most-positive-single-float
95 (single-from-bits 0 sb!vm:single-float-normal-exponent-max
96 (ldb sb!vm:single-float-significand-byte -1)))
97 (defconstant most-positive-short-float most-positive-single-float)
98 (defconstant most-negative-single-float
99 (single-from-bits 1 sb!vm:single-float-normal-exponent-max
100 (ldb sb!vm:single-float-significand-byte -1)))
101 (defconstant most-negative-short-float most-negative-single-float)
102 (defconstant most-positive-double-float
103 (double-from-bits 0 sb!vm:double-float-normal-exponent-max
104 (ldb (byte sb!vm:double-float-digits 0) -1)))
106 (defconstant most-positive-long-float most-positive-double-float)
107 #!+(and long-float x86)
108 (defconstant most-positive-long-float
109 (long-from-bits 0 sb!vm:long-float-normal-exponent-max
110 (ldb (byte sb!vm:long-float-digits 0) -1)))
111 (defconstant most-negative-double-float
112 (double-from-bits 1 sb!vm:double-float-normal-exponent-max
113 (ldb (byte sb!vm:double-float-digits 0) -1)))
115 (defconstant most-negative-long-float most-negative-double-float)
116 #!+(and long-float x86)
117 (defconstant most-negative-long-float
118 (long-from-bits 1 sb!vm:long-float-normal-exponent-max
119 (ldb (byte sb!vm:long-float-digits 0) -1)))
121 ;;; We don't want to do these DEFCONSTANTs at cross-compilation time,
122 ;;; because the cross-compilation host might not support floating
123 ;;; point infinities. Putting them inside a LET remove
124 ;;; top-level-formness, so that any EVAL-WHEN trickiness in the
125 ;;; DEFCONSTANT forms is suppressed.
127 (defconstant single-float-positive-infinity
128 (single-from-bits 0 (1+ sb!vm:single-float-normal-exponent-max) 0))
129 (defconstant short-float-positive-infinity single-float-positive-infinity)
130 (defconstant single-float-negative-infinity
131 (single-from-bits 1 (1+ sb!vm:single-float-normal-exponent-max) 0))
132 (defconstant short-float-negative-infinity single-float-negative-infinity)
133 (defconstant double-float-positive-infinity
134 (double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0))
136 (defconstant long-float-positive-infinity double-float-positive-infinity)
137 #!+(and long-float x86)
138 (defconstant long-float-positive-infinity
139 (long-from-bits 0 (1+ sb!vm:long-float-normal-exponent-max)
140 (ash sb!vm:long-float-hidden-bit 32)))
141 (defconstant double-float-negative-infinity
142 (double-from-bits 1 (1+ sb!vm:double-float-normal-exponent-max) 0))
144 (defconstant long-float-negative-infinity double-float-negative-infinity)
145 #!+(and long-float x86)
146 (defconstant long-float-negative-infinity
147 (long-from-bits 1 (1+ sb!vm:long-float-normal-exponent-max)
148 (ash sb!vm:long-float-hidden-bit 32)))
149 ) ; LET-to-suppress-possible-EVAL-WHENs
151 (defconstant single-float-epsilon
152 (single-from-bits 0 (- sb!vm:single-float-bias
153 (1- sb!vm:single-float-digits)) 1))
154 (defconstant short-float-epsilon single-float-epsilon)
155 (defconstant single-float-negative-epsilon
156 (single-from-bits 0 (- sb!vm:single-float-bias sb!vm:single-float-digits) 1))
157 (defconstant short-float-negative-epsilon single-float-negative-epsilon)
158 (defconstant double-float-epsilon
159 (double-from-bits 0 (- sb!vm:double-float-bias
160 (1- sb!vm:double-float-digits)) 1))
162 (defconstant long-float-epsilon double-float-epsilon)
163 #!+(and long-float x86)
164 (defconstant long-float-epsilon
165 (long-from-bits 0 (- sb!vm:long-float-bias (1- sb!vm:long-float-digits))
166 (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
167 (defconstant double-float-negative-epsilon
168 (double-from-bits 0 (- sb!vm:double-float-bias sb!vm:double-float-digits) 1))
170 (defconstant long-float-negative-epsilon double-float-negative-epsilon)
171 #!+(and long-float x86)
172 (defconstant long-float-negative-epsilon
173 (long-from-bits 0 (- sb!vm:long-float-bias sb!vm:long-float-digits)
174 (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
176 ;;;; float predicates and environment query
179 (declaim (maybe-inline float-denormalized-p float-infinity-p float-nan-p
180 float-trapping-nan-p))
182 (defun float-denormalized-p (x)
184 "Return true if the float X is denormalized."
185 (number-dispatch ((x float))
187 (and (zerop (ldb sb!vm:single-float-exponent-byte (single-float-bits x)))
190 (and (zerop (ldb sb!vm:double-float-exponent-byte
191 (double-float-high-bits x)))
193 #!+(and long-float x86)
195 (and (zerop (ldb sb!vm:long-float-exponent-byte (long-float-exp-bits x)))
198 (macrolet ((def-frob (name doc single double #!+(and long-float x86) long)
201 (number-dispatch ((x float))
203 (let ((bits (single-float-bits x)))
204 (and (> (ldb sb!vm:single-float-exponent-byte bits)
205 sb!vm:single-float-normal-exponent-max)
208 (let ((hi (double-float-high-bits x))
209 (lo (double-float-low-bits x)))
210 (declare (ignorable lo))
211 (and (> (ldb sb!vm:double-float-exponent-byte hi)
212 sb!vm:double-float-normal-exponent-max)
214 #!+(and long-float x86)
216 (let ((exp (long-float-exp-bits x))
217 (hi (long-float-high-bits x))
218 (lo (long-float-low-bits x)))
219 (declare (ignorable lo))
220 (and (> (ldb sb!vm:long-float-exponent-byte exp)
221 sb!vm:long-float-normal-exponent-max)
224 (def-frob float-infinity-p
225 "Return true if the float X is an infinity (+ or -)."
226 (zerop (ldb sb!vm:single-float-significand-byte bits))
227 (and (zerop (ldb sb!vm:double-float-significand-byte hi))
229 #!+(and long-float x86)
230 (and (zerop (ldb sb!vm:long-float-significand-byte hi))
233 (def-frob float-nan-p
234 "Return true if the float X is a NaN (Not a Number)."
235 (not (zerop (ldb sb!vm:single-float-significand-byte bits)))
236 (or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
238 #!+(and long-float x86)
239 (or (not (zerop (ldb sb!vm:long-float-significand-byte hi)))
242 (def-frob float-trapping-nan-p
243 "Return true if the float X is a trapping NaN (Not a Number)."
244 (zerop (logand (ldb sb!vm:single-float-significand-byte bits)
245 sb!vm:single-float-trapping-nan-bit))
246 (zerop (logand (ldb sb!vm:double-float-significand-byte hi)
247 sb!vm:double-float-trapping-nan-bit))
248 #!+(and long-float x86)
249 (zerop (logand (ldb sb!vm:long-float-significand-byte hi)
250 sb!vm:long-float-trapping-nan-bit))))
252 ;;; If denormalized, use a subfunction from INTEGER-DECODE-FLOAT to find the
253 ;;; actual exponent (and hence how denormalized it is), otherwise we just
254 ;;; return the number of digits or 0.
255 #!-sb-fluid (declaim (maybe-inline float-precision))
256 (defun float-precision (f)
258 "Returns a non-negative number of significant digits in its float argument.
259 Will be less than FLOAT-DIGITS if denormalized or zero."
260 (macrolet ((frob (digits bias decode)
262 ((float-denormalized-p f)
263 (multiple-value-bind (ignore exp) (,decode f)
264 (declare (ignore ignore))
266 (+ ,digits (1- ,digits) ,bias exp))))
269 (number-dispatch ((f float))
271 (frob sb!vm:single-float-digits sb!vm:single-float-bias
272 integer-decode-single-denorm))
274 (frob sb!vm:double-float-digits sb!vm:double-float-bias
275 integer-decode-double-denorm))
278 (frob sb!vm:long-float-digits sb!vm:long-float-bias
279 integer-decode-long-denorm)))))
281 (defun float-sign (float1 &optional (float2 (float 1 float1)))
283 "Returns a floating-point number that has the same sign as
284 float1 and, if float2 is given, has the same absolute value
286 (declare (float float1 float2))
287 (* (if (etypecase float1
288 (single-float (minusp (single-float-bits float1)))
289 (double-float (minusp (double-float-high-bits float1)))
291 (long-float (minusp (long-float-exp-bits float1))))
296 (defun float-format-digits (format)
298 ((short-float single-float) sb!vm:single-float-digits)
299 ((double-float #!-long-float long-float) sb!vm:double-float-digits)
301 (long-float sb!vm:long-float-digits)))
303 #!-sb-fluid (declaim (inline float-digits float-radix))
305 (defun float-digits (f)
306 (number-dispatch ((f float))
307 ((single-float) sb!vm:single-float-digits)
308 ((double-float) sb!vm:double-float-digits)
310 ((long-float) sb!vm:long-float-digits)))
312 (defun float-radix (x)
314 "Return (as an integer) the radix b of its floating-point argument."
315 (declare (type float x))
316 ;; ANSI says this function "should signal an error if [..] argument
317 ;; is not a float". Since X is otherwise ignored, Python doesn't
318 ;; check the type by default, so we have to do it ourself:
320 (error 'type-error :datum x :expected-type 'float))
323 ;;;; INTEGER-DECODE-FLOAT and DECODE-FLOAT
326 (declaim (maybe-inline integer-decode-single-float
327 integer-decode-double-float))
329 ;;; Handle the denormalized case of INTEGER-DECODE-FLOAT for SINGLE-FLOAT.
330 (defun integer-decode-single-denorm (x)
331 (declare (type single-float x))
332 (let* ((bits (single-float-bits (abs x)))
333 (sig (ash (ldb sb!vm:single-float-significand-byte bits) 1))
335 (declare (type (unsigned-byte 24) sig)
336 (type (integer 0 23) extra-bias))
338 (unless (zerop (logand sig sb!vm:single-float-hidden-bit))
340 (setq sig (ash sig 1))
343 (- (- sb!vm:single-float-bias)
344 sb!vm:single-float-digits
346 (if (minusp (float-sign x)) -1 1))))
348 ;;; Handle the single-float case of INTEGER-DECODE-FLOAT. If an infinity or
349 ;;; NaN, error. If a denorm, call i-d-s-DENORM to handle it.
350 (defun integer-decode-single-float (x)
351 (declare (single-float x))
352 (let* ((bits (single-float-bits (abs x)))
353 (exp (ldb sb!vm:single-float-exponent-byte bits))
354 (sig (ldb sb!vm:single-float-significand-byte bits))
355 (sign (if (minusp (float-sign x)) -1 1))
356 (biased (- exp sb!vm:single-float-bias sb!vm:single-float-digits)))
357 (declare (fixnum biased))
358 (unless (<= exp sb!vm:single-float-normal-exponent-max)
359 (error "can't decode NaN or infinity: ~S" x))
360 (cond ((and (zerop exp) (zerop sig))
361 (values 0 biased sign))
362 ((< exp sb!vm:single-float-normal-exponent-min)
363 (integer-decode-single-denorm x))
365 (values (logior sig sb!vm:single-float-hidden-bit) biased sign)))))
367 ;;; Like INTEGER-DECODE-SINGLE-DENORM, only doubly so.
368 (defun integer-decode-double-denorm (x)
369 (declare (type double-float x))
370 (let* ((high-bits (double-float-high-bits (abs x)))
371 (sig-high (ldb sb!vm:double-float-significand-byte high-bits))
372 (low-bits (double-float-low-bits x))
373 (sign (if (minusp (float-sign x)) -1 1))
374 (biased (- (- sb!vm:double-float-bias) sb!vm:double-float-digits)))
377 (extra-bias (- sb!vm:double-float-digits 33))
379 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
381 (unless (zerop (logand sig bit)) (return))
382 (setq sig (ash sig 1))
384 (values (ash sig (- sb!vm:double-float-digits 32))
385 (truly-the fixnum (- biased extra-bias))
387 (let ((sig (ash sig-high 1))
389 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
391 (unless (zerop (logand sig sb!vm:double-float-hidden-bit))
393 (setq sig (ash sig 1))
395 (values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
396 (truly-the fixnum (- biased extra-bias))
399 ;;; Like INTEGER-DECODE-SINGLE-FLOAT, only doubly so.
400 (defun integer-decode-double-float (x)
401 (declare (double-float x))
403 (hi (double-float-high-bits abs))
404 (lo (double-float-low-bits abs))
405 (exp (ldb sb!vm:double-float-exponent-byte hi))
406 (sig (ldb sb!vm:double-float-significand-byte hi))
407 (sign (if (minusp (float-sign x)) -1 1))
408 (biased (- exp sb!vm:double-float-bias sb!vm:double-float-digits)))
409 (declare (fixnum biased))
410 (unless (<= exp sb!vm:double-float-normal-exponent-max)
411 (error "Can't decode NaN or infinity: ~S." x))
412 (cond ((and (zerop exp) (zerop sig) (zerop lo))
413 (values 0 biased sign))
414 ((< exp sb!vm:double-float-normal-exponent-min)
415 (integer-decode-double-denorm x))
418 (logior (ash (logior (ldb sb!vm:double-float-significand-byte hi)
419 sb!vm:double-float-hidden-bit)
424 #!+(and long-float x86)
425 (defun integer-decode-long-denorm (x)
426 (declare (type long-float x))
427 (let* ((high-bits (long-float-high-bits (abs x)))
428 (sig-high (ldb sb!vm:long-float-significand-byte high-bits))
429 (low-bits (long-float-low-bits x))
430 (sign (if (minusp (float-sign x)) -1 1))
431 (biased (- (- sb!vm:long-float-bias) sb!vm:long-float-digits)))
434 (extra-bias (- sb!vm:long-float-digits 33))
436 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
438 (unless (zerop (logand sig bit)) (return))
439 (setq sig (ash sig 1))
441 (values (ash sig (- sb!vm:long-float-digits 32))
442 (truly-the fixnum (- biased extra-bias))
444 (let ((sig (ash sig-high 1))
446 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
448 (unless (zerop (logand sig sb!vm:long-float-hidden-bit))
450 (setq sig (ash sig 1))
452 (values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
453 (truly-the fixnum (- biased extra-bias))
456 #!+(and long-float x86)
457 (defun integer-decode-long-float (x)
458 (declare (long-float x))
459 (let* ((hi (long-float-high-bits x))
460 (lo (long-float-low-bits x))
461 (exp-bits (long-float-exp-bits x))
462 (exp (ldb sb!vm:long-float-exponent-byte exp-bits))
463 (sign (if (minusp exp-bits) -1 1))
464 (biased (- exp sb!vm:long-float-bias sb!vm:long-float-digits)))
465 (declare (fixnum biased))
466 (unless (<= exp sb!vm:long-float-normal-exponent-max)
467 (error "can't decode NaN or infinity: ~S" x))
468 (cond ((and (zerop exp) (zerop hi) (zerop lo))
469 (values 0 biased sign))
470 ((< exp sb!vm:long-float-normal-exponent-min)
471 (integer-decode-long-denorm x))
473 (values (logior (ash hi 32) lo) biased sign)))))
475 ;;; Dispatch to the correct type-specific i-d-f function.
476 (defun integer-decode-float (x)
478 "Returns three values:
479 1) an integer representation of the significand.
480 2) the exponent for the power of 2 that the significand must be multiplied
481 by to get the actual value. This differs from the DECODE-FLOAT exponent
482 by FLOAT-DIGITS, since the significand has been scaled to have all its
483 digits before the radix point.
484 3) -1 or 1 (i.e. the sign of the argument.)"
485 (number-dispatch ((x float))
487 (integer-decode-single-float x))
489 (integer-decode-double-float x))
492 (integer-decode-long-float x))))
494 #!-sb-fluid (declaim (maybe-inline decode-single-float decode-double-float))
496 ;;; Handle the denormalized case of DECODE-SINGLE-FLOAT. We call
497 ;;; INTEGER-DECODE-SINGLE-DENORM and then make the result into a float.
498 (defun decode-single-denorm (x)
499 (declare (type single-float x))
500 (multiple-value-bind (sig exp sign) (integer-decode-single-denorm x)
501 (values (make-single-float
502 (dpb sig sb!vm:single-float-significand-byte
503 (dpb sb!vm:single-float-bias
504 sb!vm:single-float-exponent-byte
506 (truly-the fixnum (+ exp sb!vm:single-float-digits))
509 ;;; Handle the single-float case of DECODE-FLOAT. If an infinity or NaN,
510 ;;; error. If a denorm, call d-s-DENORM to handle it.
511 (defun decode-single-float (x)
512 (declare (single-float x))
513 (let* ((bits (single-float-bits (abs x)))
514 (exp (ldb sb!vm:single-float-exponent-byte bits))
515 (sign (float-sign x))
516 (biased (truly-the single-float-exponent
517 (- exp sb!vm:single-float-bias))))
518 (unless (<= exp sb!vm:single-float-normal-exponent-max)
519 (error "can't decode NaN or infinity: ~S" x))
521 (values 0.0f0 biased sign))
522 ((< exp sb!vm:single-float-normal-exponent-min)
523 (decode-single-denorm x))
525 (values (make-single-float
526 (dpb sb!vm:single-float-bias
527 sb!vm:single-float-exponent-byte
531 ;;; Like DECODE-SINGLE-DENORM, only doubly so.
532 (defun decode-double-denorm (x)
533 (declare (double-float x))
534 (multiple-value-bind (sig exp sign) (integer-decode-double-denorm x)
535 (values (make-double-float
536 (dpb (logand (ash sig -32) (lognot sb!vm:double-float-hidden-bit))
537 sb!vm:double-float-significand-byte
538 (dpb sb!vm:double-float-bias
539 sb!vm:double-float-exponent-byte 0))
540 (ldb (byte 32 0) sig))
541 (truly-the fixnum (+ exp sb!vm:double-float-digits))
544 ;;; Like DECODE-SINGLE-FLOAT, only doubly so.
545 (defun decode-double-float (x)
546 (declare (double-float x))
548 (hi (double-float-high-bits abs))
549 (lo (double-float-low-bits abs))
550 (exp (ldb sb!vm:double-float-exponent-byte hi))
551 (sign (float-sign x))
552 (biased (truly-the double-float-exponent
553 (- exp sb!vm:double-float-bias))))
554 (unless (<= exp sb!vm:double-float-normal-exponent-max)
555 (error "can't decode NaN or infinity: ~S" x))
557 (values 0.0d0 biased sign))
558 ((< exp sb!vm:double-float-normal-exponent-min)
559 (decode-double-denorm x))
561 (values (make-double-float
562 (dpb sb!vm:double-float-bias
563 sb!vm:double-float-exponent-byte hi)
567 #!+(and long-float x86)
568 (defun decode-long-denorm (x)
569 (declare (long-float x))
570 (multiple-value-bind (sig exp sign) (integer-decode-long-denorm x)
571 (values (make-long-float sb!vm:long-float-bias (ash sig -32)
572 (ldb (byte 32 0) sig))
573 (truly-the fixnum (+ exp sb!vm:long-float-digits))
576 #!+(and long-float x86)
577 (defun decode-long-float (x)
578 (declare (long-float x))
579 (let* ((hi (long-float-high-bits x))
580 (lo (long-float-low-bits x))
581 (exp-bits (long-float-exp-bits x))
582 (exp (ldb sb!vm:long-float-exponent-byte exp-bits))
583 (sign (if (minusp exp-bits) -1l0 1l0))
584 (biased (truly-the long-float-exponent
585 (- exp sb!vm:long-float-bias))))
586 (unless (<= exp sb!vm:long-float-normal-exponent-max)
587 (error "can't decode NaN or infinity: ~S" x))
589 (values 0.0l0 biased sign))
590 ((< exp sb!vm:long-float-normal-exponent-min)
591 (decode-long-denorm x))
593 (values (make-long-float
594 (dpb sb!vm:long-float-bias sb!vm:long-float-exponent-byte
600 ;;; Dispatch to the appropriate type-specific function.
601 (defun decode-float (f)
603 "Returns three values:
604 1) a floating-point number representing the significand. This is always
605 between 0.5 (inclusive) and 1.0 (exclusive).
606 2) an integer representing the exponent.
607 3) -1.0 or 1.0 (i.e. the sign of the argument.)"
608 (number-dispatch ((f float))
610 (decode-single-float f))
612 (decode-double-float f))
615 (decode-long-float f))))
619 #!-sb-fluid (declaim (maybe-inline scale-single-float scale-double-float))
621 ;;; Handle float scaling where the X is denormalized or the result is
622 ;;; denormalized or underflows to 0.
623 (defun scale-float-maybe-underflow (x exp)
624 (multiple-value-bind (sig old-exp) (integer-decode-float x)
625 (let* ((digits (float-digits x))
626 (new-exp (+ exp old-exp digits
628 (single-float sb!vm:single-float-bias)
629 (double-float sb!vm:double-float-bias))))
630 (sign (if (minusp (float-sign x)) 1 0)))
634 (single-float sb!vm:single-float-normal-exponent-min)
635 (double-float sb!vm:double-float-normal-exponent-min)))
636 (when (sb!vm:current-float-trap :inexact)
637 (error 'floating-point-inexact :operation 'scale-float
638 :operands (list x exp)))
639 (when (sb!vm:current-float-trap :underflow)
640 (error 'floating-point-underflow :operation 'scale-float
641 :operands (list x exp)))
642 (let ((shift (1- new-exp)))
643 (if (< shift (- (1- digits)))
646 (single-float (single-from-bits sign 0 (ash sig shift)))
647 (double-float (double-from-bits sign 0 (ash sig shift)))))))
650 (single-float (single-from-bits sign new-exp sig))
651 (double-float (double-from-bits sign new-exp sig))))))))
653 ;;; Called when scaling a float overflows, or the original float was a
654 ;;; NaN or infinity. If overflow errors are trapped, then error,
655 ;;; otherwise return the appropriate infinity. If a NaN, signal or not
657 (defun scale-float-maybe-overflow (x exp)
659 ((float-infinity-p x)
660 ;; Infinity is infinity, no matter how small...
663 (when (and (float-trapping-nan-p x)
664 (sb!vm:current-float-trap :invalid))
665 (error 'floating-point-invalid-operation :operation 'scale-float
666 :operands (list x exp)))
669 (when (sb!vm:current-float-trap :overflow)
670 (error 'floating-point-overflow :operation 'scale-float
671 :operands (list x exp)))
672 (when (sb!vm:current-float-trap :inexact)
673 (error 'floating-point-inexact :operation 'scale-float
674 :operands (list x exp)))
677 (single-float single-float-positive-infinity)
678 (double-float double-float-positive-infinity))))))
680 ;;; Scale a single or double float, calling the correct over/underflow
682 (defun scale-single-float (x exp)
683 (declare (single-float x) (fixnum exp))
684 (let* ((bits (single-float-bits x))
685 (old-exp (ldb sb!vm:single-float-exponent-byte bits))
686 (new-exp (+ old-exp exp)))
689 ((or (< old-exp sb!vm:single-float-normal-exponent-min)
690 (< new-exp sb!vm:single-float-normal-exponent-min))
691 (scale-float-maybe-underflow x exp))
692 ((or (> old-exp sb!vm:single-float-normal-exponent-max)
693 (> new-exp sb!vm:single-float-normal-exponent-max))
694 (scale-float-maybe-overflow x exp))
696 (make-single-float (dpb new-exp
697 sb!vm:single-float-exponent-byte
699 (defun scale-double-float (x exp)
700 (declare (double-float x) (fixnum exp))
701 (let* ((hi (double-float-high-bits x))
702 (lo (double-float-low-bits x))
703 (old-exp (ldb sb!vm:double-float-exponent-byte hi))
704 (new-exp (+ old-exp exp)))
707 ((or (< old-exp sb!vm:double-float-normal-exponent-min)
708 (< new-exp sb!vm:double-float-normal-exponent-min))
709 (scale-float-maybe-underflow x exp))
710 ((or (> old-exp sb!vm:double-float-normal-exponent-max)
711 (> new-exp sb!vm:double-float-normal-exponent-max))
712 (scale-float-maybe-overflow x exp))
714 (make-double-float (dpb new-exp sb!vm:double-float-exponent-byte hi)
717 #!+(and x86 long-float)
718 (defun scale-long-float (x exp)
719 (declare (long-float x) (fixnum exp))
722 ;;; Dispatch to the correct type-specific scale-float function.
723 (defun scale-float (f ex)
725 "Returns the value (* f (expt (float 2 f) ex)), but with no unnecessary loss
726 of precision or overflow."
727 (number-dispatch ((f float))
729 (scale-single-float f ex))
731 (scale-double-float f ex))
734 (scale-long-float f ex))))
736 ;;;; converting to/from floats
738 (defun float (number &optional (other () otherp))
740 "Converts any REAL to a float. If OTHER is not provided, it returns a
741 SINGLE-FLOAT if NUMBER is not already a FLOAT. If OTHER is provided, the
742 result is the same float format as OTHER."
744 (number-dispatch ((number real) (other float))
745 (((foreach rational single-float double-float #!+long-float long-float)
746 (foreach single-float double-float #!+long-float long-float))
747 (coerce number '(dispatch-type other))))
750 (coerce number 'single-float))))
752 (macrolet ((frob (name type)
754 (number-dispatch ((x real))
755 (((foreach single-float double-float #!+long-float long-float
759 (bignum-to-float x ',type))
761 (float-ratio x ',type))))))
762 (frob %single-float single-float)
763 (frob %double-float double-float)
765 (frob %long-float long-float))
767 ;;; Convert a ratio to a float. We avoid any rounding error by doing an
768 ;;; integer division. Accuracy is important to preserve read/print
769 ;;; consistency, since this is ultimately how the reader reads a float. We
770 ;;; scale the numerator by a power of two until the division results in the
771 ;;; desired number of fraction bits, then do round-to-nearest.
772 (defun float-ratio (x format)
773 (let* ((signed-num (numerator x))
774 (plusp (plusp signed-num))
775 (num (if plusp signed-num (- signed-num)))
776 (den (denominator x))
777 (digits (float-format-digits format))
779 (declare (fixnum digits scale))
780 ;; Strip any trailing zeros from the denominator and move it into the scale
781 ;; factor (to minimize the size of the operands.)
782 (let ((den-twos (1- (integer-length (logxor den (1- den))))))
783 (declare (fixnum den-twos))
784 (decf scale den-twos)
785 (setq den (ash den (- den-twos))))
786 ;; Guess how much we need to scale by from the magnitudes of the numerator
787 ;; and denominator. We want one extra bit for a guard bit.
788 (let* ((num-len (integer-length num))
789 (den-len (integer-length den))
790 (delta (- den-len num-len))
791 (shift (1+ (the fixnum (+ delta digits))))
792 (shifted-num (ash num shift)))
793 (declare (fixnum delta shift))
795 (labels ((float-and-scale (bits)
796 (let* ((bits (ash bits -1))
797 (len (integer-length bits)))
798 (cond ((> len digits)
799 (aver (= len (the fixnum (1+ digits))))
800 (scale-float (floatit (ash bits -1)) (1+ scale)))
802 (scale-float (floatit bits) scale)))))
804 (let ((sign (if plusp 0 1)))
807 (single-from-bits sign sb!vm:single-float-bias bits))
809 (double-from-bits sign sb!vm:double-float-bias bits))
812 (long-from-bits sign sb!vm:long-float-bias bits))))))
814 (multiple-value-bind (fraction-and-guard rem)
815 (truncate shifted-num den)
816 (let ((extra (- (integer-length fraction-and-guard) digits)))
817 (declare (fixnum extra))
820 ((oddp fraction-and-guard)
824 (if (zerop (logand fraction-and-guard 2))
826 (1+ fraction-and-guard)))
827 (float-and-scale (1+ fraction-and-guard)))))
829 (return (float-and-scale fraction-and-guard)))))
830 (setq shifted-num (ash shifted-num -1))
834 These might be useful if we ever have a machine without float/integer
835 conversion hardware. For now, we'll use special ops that
836 uninterruptibly frob the rounding modes & do ieee round-to-integer.
838 ;;; The compiler compiles a call to this when we are doing %UNARY-TRUNCATE
839 ;;; and the result is known to be a fixnum. We can avoid some generic
840 ;;; arithmetic in this case.
841 (defun %unary-truncate-single-float/fixnum (x)
842 (declare (single-float x) (values fixnum))
843 (locally (declare (optimize (speed 3) (safety 0)))
844 (let* ((bits (single-float-bits x))
845 (exp (ldb sb!vm:single-float-exponent-byte bits))
846 (frac (logior (ldb sb!vm:single-float-significand-byte bits)
847 sb!vm:single-float-hidden-bit))
848 (shift (- exp sb!vm:single-float-digits sb!vm:single-float-bias)))
849 (when (> exp sb!vm:single-float-normal-exponent-max)
850 (error 'floating-point-invalid-operation :operator 'truncate
852 (if (<= shift (- sb!vm:single-float-digits))
854 (let ((res (ash frac shift)))
855 (declare (type (unsigned-byte 31) res))
860 ;;; Double-float version of this operation (see above single op).
861 (defun %unary-truncate-double-float/fixnum (x)
862 (declare (double-float x) (values fixnum))
863 (locally (declare (optimize (speed 3) (safety 0)))
864 (let* ((hi-bits (double-float-high-bits x))
865 (exp (ldb sb!vm:double-float-exponent-byte hi-bits))
866 (frac (logior (ldb sb!vm:double-float-significand-byte hi-bits)
867 sb!vm:double-float-hidden-bit))
868 (shift (- exp (- sb!vm:double-float-digits sb!vm:word-bits)
869 sb!vm:double-float-bias)))
870 (when (> exp sb!vm:double-float-normal-exponent-max)
871 (error 'floating-point-invalid-operation :operator 'truncate
873 (if (<= shift (- sb!vm:word-bits sb!vm:double-float-digits))
875 (let* ((res-hi (ash frac shift))
876 (res (if (plusp shift)
879 (ash (double-float-low-bits x)
880 (- shift sb!vm:word-bits))))
882 (declare (type (unsigned-byte 31) res-hi res))
888 ;;; This function is called when we are doing a truncate without any funky
889 ;;; divisor, i.e. converting a float or ratio to an integer. Note that we do
890 ;;; *not* return the second value of truncate, so it must be computed by the
891 ;;; caller if needed.
893 ;;; In the float case, we pick off small arguments so that compiler can use
894 ;;; special-case operations. We use an exclusive test, since (due to round-off
895 ;;; error), (float most-positive-fixnum) may be greater than
896 ;;; most-positive-fixnum.
897 (defun %unary-truncate (number)
898 (number-dispatch ((number real))
900 ((ratio) (values (truncate (numerator number) (denominator number))))
901 (((foreach single-float double-float #!+long-float long-float))
902 (if (< (float most-negative-fixnum number)
904 (float most-positive-fixnum number))
905 (truly-the fixnum (%unary-truncate number))
906 (multiple-value-bind (bits exp) (integer-decode-float number)
907 (let ((res (ash bits exp)))
912 ;;; Similar to %UNARY-TRUNCATE, but rounds to the nearest integer. If we
913 ;;; can't use the round primitive, then we do our own round-to-nearest on the
914 ;;; result of i-d-f. [Note that this rounding will really only happen with
915 ;;; double floats, since the whole single-float fraction will fit in a fixnum,
916 ;;; so all single-floats larger than most-positive-fixnum can be precisely
917 ;;; represented by an integer.]
918 (defun %unary-round (number)
919 (number-dispatch ((number real))
921 ((ratio) (values (round (numerator number) (denominator number))))
922 (((foreach single-float double-float #!+long-float long-float))
923 (if (< (float most-negative-fixnum number)
925 (float most-positive-fixnum number))
926 (truly-the fixnum (%unary-round number))
927 (multiple-value-bind (bits exp) (integer-decode-float number)
928 (let* ((shifted (ash bits exp))
929 (rounded (if (and (minusp exp)
932 (lognot (ash -1 (- exp))))
942 "RATIONAL produces a rational number for any real numeric argument. This is
943 more efficient than RATIONALIZE, but it assumes that floating-point is
944 completely accurate, giving a result that isn't as pretty."
945 (number-dispatch ((x real))
946 (((foreach single-float double-float #!+long-float long-float))
947 (multiple-value-bind (bits exp) (integer-decode-float x)
950 (let* ((int (if (minusp x) (- bits) bits))
951 (digits (float-digits x))
954 (integer-/-integer int (ash 1 (+ digits (- ex))))
955 (integer-/-integer (ash int ex) (ash 1 digits)))))))
958 (defun rationalize (x)
960 "Converts any REAL to a RATIONAL. Floats are converted to a simple rational
961 representation exploiting the assumption that floats are only accurate to
962 their precision. RATIONALIZE (and also RATIONAL) preserve the invariant:
963 (= x (float (rationalize x) x))"
964 (number-dispatch ((x real))
965 (((foreach single-float double-float #!+long-float long-float))
966 ;; Thanks to Kim Fateman, who stole this function rationalize-float from
967 ;; macsyma's rational. Macsyma'a rationalize was written by the legendary
968 ;; Gosper (rwg). Guy Steele said about Gosper, "He has been called the
969 ;; only living 17th century mathematician and is also the best pdp-10
970 ;; hacker I know." So, if you can understand or debug this code you win
972 (cond ((minusp x) (- (rationalize (- x))))
975 (let ((eps (etypecase x
976 (single-float single-float-epsilon)
977 (double-float double-float-epsilon)
979 (long-float long-float-epsilon)))
982 (do ((xx x (setq y (/ (float 1.0 x) (- xx (float a x)))))
983 (num (setq a (truncate x))
984 (+ (* (setq a (truncate y)) num) onum))
985 (den 1 (+ (* a den) oden))
988 ((and (not (zerop den))
989 (not (> (abs (/ (- x (/ (float num x)
993 (integer-/-integer num den))
994 (declare ((dispatch-type x) xx)))))))