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")
23 (eval-when (:compile-toplevel :load-toplevel :execute)
25 ;;; These functions let us create floats from bits with the significand
26 ;;; uniformly represented as an integer. This is less efficient for double
27 ;;; floats, but is more convenient when making special values, etc.
28 (defun single-from-bits (sign exp sig)
29 (declare (type bit sign) (type (unsigned-byte 24) sig)
30 (type (unsigned-byte 8) exp))
32 (dpb exp sb!vm:single-float-exponent-byte
33 (dpb sig sb!vm:single-float-significand-byte
34 (if (zerop sign) 0 -1)))))
35 (defun double-from-bits (sign exp sig)
36 (declare (type bit sign) (type (unsigned-byte 53) sig)
37 (type (unsigned-byte 11) exp))
38 (make-double-float (dpb exp sb!vm:double-float-exponent-byte
39 (dpb (ash sig -32) sb!vm:double-float-significand-byte
40 (if (zerop sign) 0 -1)))
41 (ldb (byte 32 0) sig)))
42 #!+(and long-float x86)
43 (defun long-from-bits (sign exp sig)
44 (declare (type bit sign) (type (unsigned-byte 64) sig)
45 (type (unsigned-byte 15) exp))
46 (make-long-float (logior (ash sign 15) exp)
47 (ldb (byte 32 32) sig)
48 (ldb (byte 32 0) sig)))
54 (defconstant least-positive-single-float (single-from-bits 0 0 1))
55 (defconstant least-positive-short-float least-positive-single-float)
56 (defconstant least-negative-single-float (single-from-bits 1 0 1))
57 (defconstant least-negative-short-float least-negative-single-float)
58 (defconstant least-positive-double-float (double-from-bits 0 0 1))
60 (defconstant least-positive-long-float least-positive-double-float)
61 #!+(and long-float x86)
62 (defconstant least-positive-long-float (long-from-bits 0 0 1))
63 (defconstant least-negative-double-float (double-from-bits 1 0 1))
65 (defconstant least-negative-long-float least-negative-double-float)
66 #!+(and long-float x86)
67 (defconstant least-negative-long-float (long-from-bits 1 0 1))
69 (defconstant least-positive-normalized-single-float
70 (single-from-bits 0 sb!vm:single-float-normal-exponent-min 0))
71 (defconstant least-positive-normalized-short-float
72 least-positive-normalized-single-float)
73 (defconstant least-negative-normalized-single-float
74 (single-from-bits 1 sb!vm:single-float-normal-exponent-min 0))
75 (defconstant least-negative-normalized-short-float
76 least-negative-normalized-single-float)
77 (defconstant least-positive-normalized-double-float
78 (double-from-bits 0 sb!vm:double-float-normal-exponent-min 0))
80 (defconstant least-positive-normalized-long-float
81 least-positive-normalized-double-float)
82 #!+(and long-float x86)
83 (defconstant least-positive-normalized-long-float
84 (long-from-bits 0 sb!vm:long-float-normal-exponent-min
85 (ash sb!vm:long-float-hidden-bit 32)))
86 (defconstant least-negative-normalized-double-float
87 (double-from-bits 1 sb!vm:double-float-normal-exponent-min 0))
89 (defconstant least-negative-normalized-long-float
90 least-negative-normalized-double-float)
91 #!+(and long-float x86)
92 (defconstant least-negative-normalized-long-float
93 (long-from-bits 1 sb!vm:long-float-normal-exponent-min
94 (ash sb!vm:long-float-hidden-bit 32)))
96 (defconstant most-positive-single-float
97 (single-from-bits 0 sb!vm:single-float-normal-exponent-max
98 (ldb sb!vm:single-float-significand-byte -1)))
99 (defconstant most-positive-short-float most-positive-single-float)
100 (defconstant most-negative-single-float
101 (single-from-bits 1 sb!vm:single-float-normal-exponent-max
102 (ldb sb!vm:single-float-significand-byte -1)))
103 (defconstant most-negative-short-float most-negative-single-float)
104 (defconstant most-positive-double-float
105 (double-from-bits 0 sb!vm:double-float-normal-exponent-max
106 (ldb (byte sb!vm:double-float-digits 0) -1)))
108 (defconstant most-positive-long-float most-positive-double-float)
109 #!+(and long-float x86)
110 (defconstant most-positive-long-float
111 (long-from-bits 0 sb!vm:long-float-normal-exponent-max
112 (ldb (byte sb!vm:long-float-digits 0) -1)))
113 (defconstant most-negative-double-float
114 (double-from-bits 1 sb!vm:double-float-normal-exponent-max
115 (ldb (byte sb!vm:double-float-digits 0) -1)))
117 (defconstant most-negative-long-float most-negative-double-float)
118 #!+(and long-float x86)
119 (defconstant most-negative-long-float
120 (long-from-bits 1 sb!vm:long-float-normal-exponent-max
121 (ldb (byte sb!vm:long-float-digits 0) -1)))
124 (defconstant single-float-positive-infinity
125 (single-from-bits 0 (1+ sb!vm:single-float-normal-exponent-max) 0))
127 (defconstant short-float-positive-infinity single-float-positive-infinity)
129 (defconstant single-float-negative-infinity
130 (single-from-bits 1 (1+ sb!vm:single-float-normal-exponent-max) 0))
132 (defconstant short-float-negative-infinity single-float-negative-infinity)
134 (defconstant double-float-positive-infinity
135 (double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0))
136 #!+(and sb-infinities (not long-float))
137 (defconstant long-float-positive-infinity double-float-positive-infinity)
138 #!+(and sb-infinities long-float x86)
139 (defconstant long-float-positive-infinity
140 (long-from-bits 0 (1+ sb!vm:long-float-normal-exponent-max)
141 (ash sb!vm:long-float-hidden-bit 32)))
143 (defconstant double-float-negative-infinity
144 (double-from-bits 1 (1+ sb!vm:double-float-normal-exponent-max) 0))
145 #!+(and sb-infinities (not long-float))
146 (defconstant long-float-negative-infinity double-float-negative-infinity)
147 #!+(and sb-infinities long-float x86)
148 (defconstant long-float-negative-infinity
149 (long-from-bits 1 (1+ sb!vm:long-float-normal-exponent-max)
150 (ash sb!vm:long-float-hidden-bit 32)))
152 (defconstant single-float-epsilon
153 (single-from-bits 0 (- sb!vm:single-float-bias
154 (1- sb!vm:single-float-digits)) 1))
155 (defconstant short-float-epsilon single-float-epsilon)
156 (defconstant single-float-negative-epsilon
157 (single-from-bits 0 (- sb!vm:single-float-bias sb!vm:single-float-digits) 1))
158 (defconstant short-float-negative-epsilon single-float-negative-epsilon)
159 (defconstant double-float-epsilon
160 (double-from-bits 0 (- sb!vm:double-float-bias
161 (1- sb!vm:double-float-digits)) 1))
163 (defconstant long-float-epsilon double-float-epsilon)
164 #!+(and long-float x86)
165 (defconstant long-float-epsilon
166 (long-from-bits 0 (- sb!vm:long-float-bias (1- sb!vm:long-float-digits))
167 (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
168 (defconstant double-float-negative-epsilon
169 (double-from-bits 0 (- sb!vm:double-float-bias sb!vm:double-float-digits) 1))
171 (defconstant long-float-negative-epsilon double-float-negative-epsilon)
172 #!+(and long-float x86)
173 (defconstant long-float-negative-epsilon
174 (long-from-bits 0 (- sb!vm:long-float-bias sb!vm:long-float-digits)
175 (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
177 ;;;; float predicates and environment query
180 (declaim (maybe-inline float-denormalized-p float-infinity-p float-nan-p
181 float-trapping-nan-p))
183 (defun float-denormalized-p (x)
185 "Return true if the float X is denormalized."
186 (number-dispatch ((x float))
188 (and (zerop (ldb sb!vm:single-float-exponent-byte (single-float-bits x)))
191 (and (zerop (ldb sb!vm:double-float-exponent-byte
192 (double-float-high-bits x)))
194 #!+(and long-float x86)
196 (and (zerop (ldb sb!vm:long-float-exponent-byte (long-float-exp-bits x)))
199 (macrolet ((def-frob (name doc single double #!+(and long-float x86) long)
202 (number-dispatch ((x float))
204 (let ((bits (single-float-bits x)))
205 (and (> (ldb sb!vm:single-float-exponent-byte bits)
206 sb!vm:single-float-normal-exponent-max)
209 (let ((hi (double-float-high-bits x))
210 (lo (double-float-low-bits x)))
211 (declare (ignorable lo))
212 (and (> (ldb sb!vm:double-float-exponent-byte hi)
213 sb!vm:double-float-normal-exponent-max)
215 #!+(and long-float x86)
217 (let ((exp (long-float-exp-bits x))
218 (hi (long-float-high-bits x))
219 (lo (long-float-low-bits x)))
220 (declare (ignorable lo))
221 (and (> (ldb sb!vm:long-float-exponent-byte exp)
222 sb!vm:long-float-normal-exponent-max)
225 (def-frob float-infinity-p
226 "Return true if the float X is an infinity (+ or -)."
227 (zerop (ldb sb!vm:single-float-significand-byte bits))
228 (and (zerop (ldb sb!vm:double-float-significand-byte hi))
230 #!+(and long-float x86)
231 (and (zerop (ldb sb!vm:long-float-significand-byte hi))
234 (def-frob float-nan-p
235 "Return true if the float X is a NaN (Not a Number)."
236 (not (zerop (ldb sb!vm:single-float-significand-byte bits)))
237 (or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
239 #!+(and long-float x86)
240 (or (not (zerop (ldb sb!vm:long-float-significand-byte hi)))
243 (def-frob float-trapping-nan-p
244 "Return true if the float X is a trapping NaN (Not a Number)."
245 (zerop (logand (ldb sb!vm:single-float-significand-byte bits)
246 sb!vm:single-float-trapping-nan-bit))
247 (zerop (logand (ldb sb!vm:double-float-significand-byte hi)
248 sb!vm:double-float-trapping-nan-bit))
249 #!+(and long-float x86)
250 (zerop (logand (ldb sb!vm:long-float-significand-byte hi)
251 sb!vm:long-float-trapping-nan-bit))))
253 ;;; If denormalized, use a subfunction from INTEGER-DECODE-FLOAT to find the
254 ;;; actual exponent (and hence how denormalized it is), otherwise we just
255 ;;; return the number of digits or 0.
256 #!-sb-fluid (declaim (maybe-inline float-precision))
257 (defun float-precision (f)
259 "Returns a non-negative number of significant digits in its float argument.
260 Will be less than FLOAT-DIGITS if denormalized or zero."
261 (macrolet ((frob (digits bias decode)
263 ((float-denormalized-p f)
264 (multiple-value-bind (ignore exp) (,decode f)
265 (declare (ignore ignore))
267 (+ ,digits (1- ,digits) ,bias exp))))
270 (number-dispatch ((f float))
272 (frob sb!vm:single-float-digits sb!vm:single-float-bias
273 integer-decode-single-denorm))
275 (frob sb!vm:double-float-digits sb!vm:double-float-bias
276 integer-decode-double-denorm))
279 (frob sb!vm:long-float-digits sb!vm:long-float-bias
280 integer-decode-long-denorm)))))
282 (defun float-sign (float1 &optional (float2 (float 1 float1)))
284 "Returns a floating-point number that has the same sign as
285 float1 and, if float2 is given, has the same absolute value
287 (declare (float float1 float2))
288 (* (if (etypecase float1
289 (single-float (minusp (single-float-bits float1)))
290 (double-float (minusp (double-float-high-bits float1)))
292 (long-float (minusp (long-float-exp-bits float1))))
297 (defun float-format-digits (format)
299 ((short-float single-float) sb!vm:single-float-digits)
300 ((double-float #!-long-float long-float) sb!vm:double-float-digits)
302 (long-float sb!vm:long-float-digits)))
304 #!-sb-fluid (declaim (inline float-digits float-radix))
306 (defun float-digits (f)
307 (number-dispatch ((f float))
308 ((single-float) sb!vm:single-float-digits)
309 ((double-float) sb!vm:double-float-digits)
311 ((long-float) sb!vm:long-float-digits)))
313 (defun float-radix (x)
315 "Returns (as an integer) the radix b of its floating-point
317 (declare (type float x) (ignore x))
320 ;;;; INTEGER-DECODE-FLOAT and DECODE-FLOAT
323 (declaim (maybe-inline integer-decode-single-float
324 integer-decode-double-float))
326 ;;; Handle the denormalized case of INTEGER-DECODE-FLOAT for SINGLE-FLOAT.
327 (defun integer-decode-single-denorm (x)
328 (declare (type single-float x))
329 (let* ((bits (single-float-bits (abs x)))
330 (sig (ash (ldb sb!vm:single-float-significand-byte bits) 1))
332 (declare (type (unsigned-byte 24) sig)
333 (type (integer 0 23) extra-bias))
335 (unless (zerop (logand sig sb!vm:single-float-hidden-bit))
337 (setq sig (ash sig 1))
340 (- (- sb!vm:single-float-bias)
341 sb!vm:single-float-digits
343 (if (minusp (float-sign x)) -1 1))))
345 ;;; Handle the single-float case of INTEGER-DECODE-FLOAT. If an infinity or
346 ;;; NaN, error. If a denorm, call i-d-s-DENORM to handle it.
347 (defun integer-decode-single-float (x)
348 (declare (single-float x))
349 (let* ((bits (single-float-bits (abs x)))
350 (exp (ldb sb!vm:single-float-exponent-byte bits))
351 (sig (ldb sb!vm:single-float-significand-byte bits))
352 (sign (if (minusp (float-sign x)) -1 1))
353 (biased (- exp sb!vm:single-float-bias sb!vm:single-float-digits)))
354 (declare (fixnum biased))
355 (unless (<= exp sb!vm:single-float-normal-exponent-max)
356 (error "can't decode NaN or infinity: ~S" x))
357 (cond ((and (zerop exp) (zerop sig))
358 (values 0 biased sign))
359 ((< exp sb!vm:single-float-normal-exponent-min)
360 (integer-decode-single-denorm x))
362 (values (logior sig sb!vm:single-float-hidden-bit) biased sign)))))
364 ;;; Like INTEGER-DECODE-SINGLE-DENORM, only doubly so.
365 (defun integer-decode-double-denorm (x)
366 (declare (type double-float x))
367 (let* ((high-bits (double-float-high-bits (abs x)))
368 (sig-high (ldb sb!vm:double-float-significand-byte high-bits))
369 (low-bits (double-float-low-bits x))
370 (sign (if (minusp (float-sign x)) -1 1))
371 (biased (- (- sb!vm:double-float-bias) sb!vm:double-float-digits)))
374 (extra-bias (- sb!vm:double-float-digits 33))
376 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
378 (unless (zerop (logand sig bit)) (return))
379 (setq sig (ash sig 1))
381 (values (ash sig (- sb!vm:double-float-digits 32))
382 (truly-the fixnum (- biased extra-bias))
384 (let ((sig (ash sig-high 1))
386 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
388 (unless (zerop (logand sig sb!vm:double-float-hidden-bit))
390 (setq sig (ash sig 1))
392 (values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
393 (truly-the fixnum (- biased extra-bias))
396 ;;; Like INTEGER-DECODE-SINGLE-FLOAT, only doubly so.
397 (defun integer-decode-double-float (x)
398 (declare (double-float x))
400 (hi (double-float-high-bits abs))
401 (lo (double-float-low-bits abs))
402 (exp (ldb sb!vm:double-float-exponent-byte hi))
403 (sig (ldb sb!vm:double-float-significand-byte hi))
404 (sign (if (minusp (float-sign x)) -1 1))
405 (biased (- exp sb!vm:double-float-bias sb!vm:double-float-digits)))
406 (declare (fixnum biased))
407 (unless (<= exp sb!vm:double-float-normal-exponent-max)
408 (error "Can't decode NaN or infinity: ~S." x))
409 (cond ((and (zerop exp) (zerop sig) (zerop lo))
410 (values 0 biased sign))
411 ((< exp sb!vm:double-float-normal-exponent-min)
412 (integer-decode-double-denorm x))
415 (logior (ash (logior (ldb sb!vm:double-float-significand-byte hi)
416 sb!vm:double-float-hidden-bit)
421 #!+(and long-float x86)
422 (defun integer-decode-long-denorm (x)
423 (declare (type long-float x))
424 (let* ((high-bits (long-float-high-bits (abs x)))
425 (sig-high (ldb sb!vm:long-float-significand-byte high-bits))
426 (low-bits (long-float-low-bits x))
427 (sign (if (minusp (float-sign x)) -1 1))
428 (biased (- (- sb!vm:long-float-bias) sb!vm:long-float-digits)))
431 (extra-bias (- sb!vm:long-float-digits 33))
433 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
435 (unless (zerop (logand sig bit)) (return))
436 (setq sig (ash sig 1))
438 (values (ash sig (- sb!vm:long-float-digits 32))
439 (truly-the fixnum (- biased extra-bias))
441 (let ((sig (ash sig-high 1))
443 (declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
445 (unless (zerop (logand sig sb!vm:long-float-hidden-bit))
447 (setq sig (ash sig 1))
449 (values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
450 (truly-the fixnum (- biased extra-bias))
453 #!+(and long-float x86)
454 (defun integer-decode-long-float (x)
455 (declare (long-float x))
456 (let* ((hi (long-float-high-bits x))
457 (lo (long-float-low-bits x))
458 (exp-bits (long-float-exp-bits x))
459 (exp (ldb sb!vm:long-float-exponent-byte exp-bits))
460 (sign (if (minusp exp-bits) -1 1))
461 (biased (- exp sb!vm:long-float-bias sb!vm:long-float-digits)))
462 (declare (fixnum biased))
463 (unless (<= exp sb!vm:long-float-normal-exponent-max)
464 (error "can't decode NaN or infinity: ~S" x))
465 (cond ((and (zerop exp) (zerop hi) (zerop lo))
466 (values 0 biased sign))
467 ((< exp sb!vm:long-float-normal-exponent-min)
468 (integer-decode-long-denorm x))
470 (values (logior (ash hi 32) lo) biased sign)))))
472 ;;; Dispatch to the correct type-specific i-d-f function.
473 (defun integer-decode-float (x)
475 "Returns three values:
476 1) an integer representation of the significand.
477 2) the exponent for the power of 2 that the significand must be multiplied
478 by to get the actual value. This differs from the DECODE-FLOAT exponent
479 by FLOAT-DIGITS, since the significand has been scaled to have all its
480 digits before the radix point.
481 3) -1 or 1 (i.e. the sign of the argument.)"
482 (number-dispatch ((x float))
484 (integer-decode-single-float x))
486 (integer-decode-double-float x))
489 (integer-decode-long-float x))))
491 #!-sb-fluid (declaim (maybe-inline decode-single-float decode-double-float))
493 ;;; Handle the denormalized case of DECODE-SINGLE-FLOAT. We call
494 ;;; INTEGER-DECODE-SINGLE-DENORM and then make the result into a float.
495 (defun decode-single-denorm (x)
496 (declare (type single-float x))
497 (multiple-value-bind (sig exp sign) (integer-decode-single-denorm x)
498 (values (make-single-float
499 (dpb sig sb!vm:single-float-significand-byte
500 (dpb sb!vm:single-float-bias
501 sb!vm:single-float-exponent-byte
503 (truly-the fixnum (+ exp sb!vm:single-float-digits))
506 ;;; Handle the single-float case of DECODE-FLOAT. If an infinity or NaN,
507 ;;; error. If a denorm, call d-s-DENORM to handle it.
508 (defun decode-single-float (x)
509 (declare (single-float x))
510 (let* ((bits (single-float-bits (abs x)))
511 (exp (ldb sb!vm:single-float-exponent-byte bits))
512 (sign (float-sign x))
513 (biased (truly-the single-float-exponent
514 (- exp sb!vm:single-float-bias))))
515 (unless (<= exp sb!vm:single-float-normal-exponent-max)
516 (error "can't decode NaN or infinity: ~S" x))
518 (values 0.0f0 biased sign))
519 ((< exp sb!vm:single-float-normal-exponent-min)
520 (decode-single-denorm x))
522 (values (make-single-float
523 (dpb sb!vm:single-float-bias
524 sb!vm:single-float-exponent-byte
528 ;;; Like DECODE-SINGLE-DENORM, only doubly so.
529 (defun decode-double-denorm (x)
530 (declare (double-float x))
531 (multiple-value-bind (sig exp sign) (integer-decode-double-denorm x)
532 (values (make-double-float
533 (dpb (logand (ash sig -32) (lognot sb!vm:double-float-hidden-bit))
534 sb!vm:double-float-significand-byte
535 (dpb sb!vm:double-float-bias
536 sb!vm:double-float-exponent-byte 0))
537 (ldb (byte 32 0) sig))
538 (truly-the fixnum (+ exp sb!vm:double-float-digits))
541 ;;; Like DECODE-SINGLE-FLOAT, only doubly so.
542 (defun decode-double-float (x)
543 (declare (double-float x))
545 (hi (double-float-high-bits abs))
546 (lo (double-float-low-bits abs))
547 (exp (ldb sb!vm:double-float-exponent-byte hi))
548 (sign (float-sign x))
549 (biased (truly-the double-float-exponent
550 (- exp sb!vm:double-float-bias))))
551 (unless (<= exp sb!vm:double-float-normal-exponent-max)
552 (error "can't decode NaN or infinity: ~S" x))
554 (values 0.0d0 biased sign))
555 ((< exp sb!vm:double-float-normal-exponent-min)
556 (decode-double-denorm x))
558 (values (make-double-float
559 (dpb sb!vm:double-float-bias
560 sb!vm:double-float-exponent-byte hi)
564 #!+(and long-float x86)
565 (defun decode-long-denorm (x)
566 (declare (long-float x))
567 (multiple-value-bind (sig exp sign) (integer-decode-long-denorm x)
568 (values (make-long-float sb!vm:long-float-bias (ash sig -32)
569 (ldb (byte 32 0) sig))
570 (truly-the fixnum (+ exp sb!vm:long-float-digits))
573 #!+(and long-float x86)
574 (defun decode-long-float (x)
575 (declare (long-float x))
576 (let* ((hi (long-float-high-bits x))
577 (lo (long-float-low-bits x))
578 (exp-bits (long-float-exp-bits x))
579 (exp (ldb sb!vm:long-float-exponent-byte exp-bits))
580 (sign (if (minusp exp-bits) -1l0 1l0))
581 (biased (truly-the long-float-exponent
582 (- exp sb!vm:long-float-bias))))
583 (unless (<= exp sb!vm:long-float-normal-exponent-max)
584 (error "can't decode NaN or infinity: ~S" x))
586 (values 0.0l0 biased sign))
587 ((< exp sb!vm:long-float-normal-exponent-min)
588 (decode-long-denorm x))
590 (values (make-long-float
591 (dpb sb!vm:long-float-bias sb!vm:long-float-exponent-byte
597 ;;; Dispatch to the appropriate type-specific function.
598 (defun decode-float (f)
600 "Returns three values:
601 1) a floating-point number representing the significand. This is always
602 between 0.5 (inclusive) and 1.0 (exclusive).
603 2) an integer representing the exponent.
604 3) -1.0 or 1.0 (i.e. the sign of the argument.)"
605 (number-dispatch ((f float))
607 (decode-single-float f))
609 (decode-double-float f))
612 (decode-long-float f))))
616 #!-sb-fluid (declaim (maybe-inline scale-single-float scale-double-float))
618 ;;; Handle float scaling where the X is denormalized or the result is
619 ;;; denormalized or underflows to 0.
620 (defun scale-float-maybe-underflow (x exp)
621 (multiple-value-bind (sig old-exp) (integer-decode-float x)
622 (let* ((digits (float-digits x))
623 (new-exp (+ exp old-exp digits
625 (single-float sb!vm:single-float-bias)
626 (double-float sb!vm:double-float-bias))))
627 (sign (if (minusp (float-sign x)) 1 0)))
631 (single-float sb!vm:single-float-normal-exponent-min)
632 (double-float sb!vm:double-float-normal-exponent-min)))
633 (when (sb!vm:current-float-trap :inexact)
634 (error 'floating-point-inexact :operation 'scale-float
635 :operands (list x exp)))
636 (when (sb!vm:current-float-trap :underflow)
637 (error 'floating-point-underflow :operation 'scale-float
638 :operands (list x exp)))
639 (let ((shift (1- new-exp)))
640 (if (< shift (- (1- digits)))
643 (single-float (single-from-bits sign 0 (ash sig shift)))
644 (double-float (double-from-bits sign 0 (ash sig shift)))))))
647 (single-float (single-from-bits sign new-exp sig))
648 (double-float (double-from-bits sign new-exp sig))))))))
650 ;;; Called when scaling a float overflows, or the original float was a NaN
651 ;;; or infinity. If overflow errors are trapped, then error, otherwise return
652 ;;; the appropriate infinity. If a NaN, signal or not as appropriate.
653 (defun scale-float-maybe-overflow (x exp)
655 ((float-infinity-p x)
656 ;; Infinity is infinity, no matter how small...
659 (when (and (float-trapping-nan-p x)
660 (sb!vm:current-float-trap :invalid))
661 (error 'floating-point-invalid-operation :operation 'scale-float
662 :operands (list x exp)))
665 (when (sb!vm:current-float-trap :overflow)
666 (error 'floating-point-overflow :operation 'scale-float
667 :operands (list x exp)))
668 (when (sb!vm:current-float-trap :inexact)
669 (error 'floating-point-inexact :operation 'scale-float
670 :operands (list x exp)))
671 (infinite (* (float-sign x)
673 (single-float single-float-positive-infinity)
674 (double-float double-float-positive-infinity)))))))
676 ;;; Scale a single or double float, calling the correct over/underflow
678 (defun scale-single-float (x exp)
679 (declare (single-float x) (fixnum exp))
680 (let* ((bits (single-float-bits x))
681 (old-exp (ldb sb!vm:single-float-exponent-byte bits))
682 (new-exp (+ old-exp exp)))
685 ((or (< old-exp sb!vm:single-float-normal-exponent-min)
686 (< new-exp sb!vm:single-float-normal-exponent-min))
687 (scale-float-maybe-underflow x exp))
688 ((or (> old-exp sb!vm:single-float-normal-exponent-max)
689 (> new-exp sb!vm:single-float-normal-exponent-max))
690 (scale-float-maybe-overflow x exp))
692 (make-single-float (dpb new-exp
693 sb!vm:single-float-exponent-byte
695 (defun scale-double-float (x exp)
696 (declare (double-float x) (fixnum exp))
697 (let* ((hi (double-float-high-bits x))
698 (lo (double-float-low-bits x))
699 (old-exp (ldb sb!vm:double-float-exponent-byte hi))
700 (new-exp (+ old-exp exp)))
703 ((or (< old-exp sb!vm:double-float-normal-exponent-min)
704 (< new-exp sb!vm:double-float-normal-exponent-min))
705 (scale-float-maybe-underflow x exp))
706 ((or (> old-exp sb!vm:double-float-normal-exponent-max)
707 (> new-exp sb!vm:double-float-normal-exponent-max))
708 (scale-float-maybe-overflow x exp))
710 (make-double-float (dpb new-exp sb!vm:double-float-exponent-byte hi)
713 #!+(and x86 long-float)
714 (defun scale-long-float (x exp)
715 (declare (long-float x) (fixnum exp))
718 ;;; Dispatch to the correct type-specific scale-float function.
719 (defun scale-float (f ex)
721 "Returns the value (* f (expt (float 2 f) ex)), but with no unnecessary loss
722 of precision or overflow."
723 (number-dispatch ((f float))
725 (scale-single-float f ex))
727 (scale-double-float f ex))
730 (scale-long-float f ex))))
732 ;;;; converting to/from floats
734 (defun float (number &optional (other () otherp))
736 "Converts any REAL to a float. If OTHER is not provided, it returns a
737 SINGLE-FLOAT if NUMBER is not already a FLOAT. If OTHER is provided, the
738 result is the same float format as OTHER."
740 (number-dispatch ((number real) (other float))
741 (((foreach rational single-float double-float #!+long-float long-float)
742 (foreach single-float double-float #!+long-float long-float))
743 (coerce number '(dispatch-type other))))
746 (coerce number 'single-float))))
748 (macrolet ((frob (name type)
750 (number-dispatch ((x real))
751 (((foreach single-float double-float #!+long-float long-float
755 (bignum-to-float x ',type))
757 (float-ratio x ',type))))))
758 (frob %single-float single-float)
759 (frob %double-float double-float)
761 (frob %long-float long-float))
763 ;;; Convert a ratio to a float. We avoid any rounding error by doing an
764 ;;; integer division. Accuracy is important to preserve read/print
765 ;;; consistency, since this is ultimately how the reader reads a float. We
766 ;;; scale the numerator by a power of two until the division results in the
767 ;;; desired number of fraction bits, then do round-to-nearest.
768 (defun float-ratio (x format)
769 (let* ((signed-num (numerator x))
770 (plusp (plusp signed-num))
771 (num (if plusp signed-num (- signed-num)))
772 (den (denominator x))
773 (digits (float-format-digits format))
775 (declare (fixnum digits scale))
776 ;; Strip any trailing zeros from the denominator and move it into the scale
777 ;; factor (to minimize the size of the operands.)
778 (let ((den-twos (1- (integer-length (logxor den (1- den))))))
779 (declare (fixnum den-twos))
780 (decf scale den-twos)
781 (setq den (ash den (- den-twos))))
782 ;; Guess how much we need to scale by from the magnitudes of the numerator
783 ;; and denominator. We want one extra bit for a guard bit.
784 (let* ((num-len (integer-length num))
785 (den-len (integer-length den))
786 (delta (- den-len num-len))
787 (shift (1+ (the fixnum (+ delta digits))))
788 (shifted-num (ash num shift)))
789 (declare (fixnum delta shift))
791 (labels ((float-and-scale (bits)
792 (let* ((bits (ash bits -1))
793 (len (integer-length bits)))
794 (cond ((> len digits)
795 (assert (= len (the fixnum (1+ digits))))
796 (scale-float (floatit (ash bits -1)) (1+ scale)))
798 (scale-float (floatit bits) scale)))))
800 (let ((sign (if plusp 0 1)))
803 (single-from-bits sign sb!vm:single-float-bias bits))
805 (double-from-bits sign sb!vm:double-float-bias bits))
808 (long-from-bits sign sb!vm:long-float-bias bits))))))
810 (multiple-value-bind (fraction-and-guard rem)
811 (truncate shifted-num den)
812 (let ((extra (- (integer-length fraction-and-guard) digits)))
813 (declare (fixnum extra))
815 (assert (> extra 1)))
816 ((oddp fraction-and-guard)
820 (if (zerop (logand fraction-and-guard 2))
822 (1+ fraction-and-guard)))
823 (float-and-scale (1+ fraction-and-guard)))))
825 (return (float-and-scale fraction-and-guard)))))
826 (setq shifted-num (ash shifted-num -1))
830 These might be useful if we ever have a machine w/o float/integer conversion
831 hardware. For now, we'll use special ops that uninterruptibly frob the
832 rounding modes & do ieee round-to-integer.
834 ;;; The compiler compiles a call to this when we are doing %UNARY-TRUNCATE
835 ;;; and the result is known to be a fixnum. We can avoid some generic
836 ;;; arithmetic in this case.
837 (defun %unary-truncate-single-float/fixnum (x)
838 (declare (single-float x) (values fixnum))
839 (locally (declare (optimize (speed 3) (safety 0)))
840 (let* ((bits (single-float-bits x))
841 (exp (ldb sb!vm:single-float-exponent-byte bits))
842 (frac (logior (ldb sb!vm:single-float-significand-byte bits)
843 sb!vm:single-float-hidden-bit))
844 (shift (- exp sb!vm:single-float-digits sb!vm:single-float-bias)))
845 (when (> exp sb!vm:single-float-normal-exponent-max)
846 (error 'floating-point-invalid-operation :operator 'truncate
848 (if (<= shift (- sb!vm:single-float-digits))
850 (let ((res (ash frac shift)))
851 (declare (type (unsigned-byte 31) res))
856 ;;; Double-float version of this operation (see above single op).
857 (defun %unary-truncate-double-float/fixnum (x)
858 (declare (double-float x) (values fixnum))
859 (locally (declare (optimize (speed 3) (safety 0)))
860 (let* ((hi-bits (double-float-high-bits x))
861 (exp (ldb sb!vm:double-float-exponent-byte hi-bits))
862 (frac (logior (ldb sb!vm:double-float-significand-byte hi-bits)
863 sb!vm:double-float-hidden-bit))
864 (shift (- exp (- sb!vm:double-float-digits sb!vm:word-bits)
865 sb!vm:double-float-bias)))
866 (when (> exp sb!vm:double-float-normal-exponent-max)
867 (error 'floating-point-invalid-operation :operator 'truncate
869 (if (<= shift (- sb!vm:word-bits sb!vm:double-float-digits))
871 (let* ((res-hi (ash frac shift))
872 (res (if (plusp shift)
875 (ash (double-float-low-bits x)
876 (- shift sb!vm:word-bits))))
878 (declare (type (unsigned-byte 31) res-hi res))
884 ;;; This function is called when we are doing a truncate without any funky
885 ;;; divisor, i.e. converting a float or ratio to an integer. Note that we do
886 ;;; *not* return the second value of truncate, so it must be computed by the
887 ;;; caller if needed.
889 ;;; In the float case, we pick off small arguments so that compiler can use
890 ;;; special-case operations. We use an exclusive test, since (due to round-off
891 ;;; error), (float most-positive-fixnum) may be greater than
892 ;;; most-positive-fixnum.
893 (defun %unary-truncate (number)
894 (number-dispatch ((number real))
896 ((ratio) (values (truncate (numerator number) (denominator number))))
897 (((foreach single-float double-float #!+long-float long-float))
898 (if (< (float most-negative-fixnum number)
900 (float most-positive-fixnum number))
901 (truly-the fixnum (%unary-truncate number))
902 (multiple-value-bind (bits exp) (integer-decode-float number)
903 (let ((res (ash bits exp)))
908 ;;; Similar to %UNARY-TRUNCATE, but rounds to the nearest integer. If we
909 ;;; can't use the round primitive, then we do our own round-to-nearest on the
910 ;;; result of i-d-f. [Note that this rounding will really only happen with
911 ;;; double floats, since the whole single-float fraction will fit in a fixnum,
912 ;;; so all single-floats larger than most-positive-fixnum can be precisely
913 ;;; represented by an integer.]
914 (defun %unary-round (number)
915 (number-dispatch ((number real))
917 ((ratio) (values (round (numerator number) (denominator number))))
918 (((foreach single-float double-float #!+long-float long-float))
919 (if (< (float most-negative-fixnum number)
921 (float most-positive-fixnum number))
922 (truly-the fixnum (%unary-round number))
923 (multiple-value-bind (bits exp) (integer-decode-float number)
924 (let* ((shifted (ash bits exp))
925 (rounded (if (and (minusp exp)
928 (lognot (ash -1 (- exp))))
938 "RATIONAL produces a rational number for any real numeric argument. This is
939 more efficient than RATIONALIZE, but it assumes that floating-point is
940 completely accurate, giving a result that isn't as pretty."
941 (number-dispatch ((x real))
942 (((foreach single-float double-float #!+long-float long-float))
943 (multiple-value-bind (bits exp) (integer-decode-float x)
946 (let* ((int (if (minusp x) (- bits) bits))
947 (digits (float-digits x))
950 (integer-/-integer int (ash 1 (+ digits (- ex))))
951 (integer-/-integer (ash int ex) (ash 1 digits)))))))
954 (defun rationalize (x)
956 "Converts any REAL to a RATIONAL. Floats are converted to a simple rational
957 representation exploiting the assumption that floats are only accurate to
958 their precision. RATIONALIZE (and also RATIONAL) preserve the invariant:
959 (= x (float (rationalize x) x))"
960 (number-dispatch ((x real))
961 (((foreach single-float double-float #!+long-float long-float))
962 ;; Thanks to Kim Fateman, who stole this function rationalize-float from
963 ;; macsyma's rational. Macsyma'a rationalize was written by the legendary
964 ;; Gosper (rwg). Guy Steele said about Gosper, "He has been called the
965 ;; only living 17th century mathematician and is also the best pdp-10
966 ;; hacker I know." So, if you can understand or debug this code you win
968 (cond ((minusp x) (- (rationalize (- x))))
971 (let ((eps (etypecase x
972 (single-float single-float-epsilon)
973 (double-float double-float-epsilon)
975 (long-float long-float-epsilon)))
978 (do ((xx x (setq y (/ (float 1.0 x) (- xx (float a x)))))
979 (num (setq a (truncate x))
980 (+ (* (setq a (truncate y)) num) onum))
981 (den 1 (+ (* a den) oden))
984 ((and (not (zerop den))
985 (not (> (abs (/ (- x (/ (float num x)
989 (integer-/-integer num den))
990 (declare ((dispatch-type x) xx)))))))