\f
;;;; miscellaneous constants, utility functions, and macros
-(defconstant pi 3.14159265358979323846264338327950288419716939937511L0)
-;(defconstant e 2.71828182845904523536028747135266249775724709369996L0)
+(defconstant pi
+ #!+long-float 3.14159265358979323846264338327950288419716939937511l0
+ #!-long-float 3.14159265358979323846264338327950288419716939937511d0)
;;; Make these INLINE, since the call to C is at least as compact as a
;;; Lisp call, and saves number consing to boot.
(sb!xc:defmacro def-math-rtn (name num-args)
(let ((function (symbolicate "%" (string-upcase name))))
`(progn
- (proclaim '(inline ,function))
- (sb!alien:def-alien-routine (,name ,function) double-float
+ (declaim (inline ,function))
+ (sb!alien:define-alien-routine (,name ,function) double-float
,@(let ((results nil))
(dotimes (i num-args (nreverse results))
(push (list (intern (format nil "ARG-~D" i))
) ; EVAL-WHEN
\f
-;;;; stubs for the Unix math library
+#!+x86 ;; for constant folding
+(macrolet ((def (name ll)
+ `(defun ,name ,ll (,name ,@ll))))
+ (def %atan2 (x y))
+ (def %atan (x))
+ (def %tan-quick (x))
+ (def %cos-quick (x))
+ (def %sin-quick (x))
+ (def %sqrt (x))
+ (def %log (x))
+ (def %exp (x)))
-;;; Please refer to the Unix man pages for details about these routines.
+;;;; stubs for the Unix math library
+;;;;
+;;;; Many of these are unnecessary on the X86 because they're built
+;;;; into the FPU.
;;; trigonometric
#!-x86 (def-math-rtn "sin" 1)
;;; INTEXP -- Handle the rational base, integer power case.
-;;; FIXME: As long as the system dies on stack overflow or memory
-;;; exhaustion, it seems reasonable to have this, but its default
-;;; should be NIL, and when it's NIL, anything should be accepted.
-(defparameter *intexp-maximum-exponent* 10000)
+(declaim (type (or integer null) *intexp-maximum-exponent*))
+(defparameter *intexp-maximum-exponent* nil)
;;; This function precisely calculates base raised to an integral
;;; power. It separates the cases by the sign of power, for efficiency
;;; a positive integer. Values of power are calculated as positive
;;; integers, and inverted if negative.
(defun intexp (base power)
- (when (> (abs power) *intexp-maximum-exponent*)
- ;; FIXME: should be ordinary error, not CERROR. (Once we set the
- ;; default for the variable to NIL, the un-continuable error will
- ;; be less obnoxious.)
- (cerror "Continue with calculation."
- "The absolute value of ~S exceeds ~S."
- power '*intexp-maximum-exponent* base power))
+ (when (and *intexp-maximum-exponent*
+ (> (abs power) *intexp-maximum-exponent*))
+ (error "The absolute value of ~S exceeds ~S."
+ power '*intexp-maximum-exponent*))
(cond ((minusp power)
(/ (intexp base (- power))))
((eql base 2)
#!+sb-doc
"Return BASE raised to the POWER."
(if (zerop power)
- (1+ (* base power))
+ (let ((result (1+ (* base power))))
+ (if (and (floatp result) (float-nan-p result))
+ (float 1 result)
+ result))
(labels (;; determine if the double float is an integer.
;; 0 - not an integer
;; 1 - an odd int
(* base power)
(exp (* power (log base)))))))))
+;;; FIXME: Maybe rename this so that it's clearer that it only works
+;;; on integers?
+(defun log2 (x)
+ (declare (type integer x))
+ ;; CMUCL comment:
+ ;;
+ ;; Write x = 2^n*f where 1/2 < f <= 1. Then log2(x) = n +
+ ;; log2(f). So we grab the top few bits of x and scale that
+ ;; appropriately, take the log of it and add it to n.
+ ;;
+ ;; Motivated by an attempt to get LOG to work better on bignums.
+ (let ((n (integer-length x)))
+ (if (< n sb!vm:double-float-digits)
+ (log (coerce x 'double-float) 2.0d0)
+ (let ((f (ldb (byte sb!vm:double-float-digits
+ (- n sb!vm:double-float-digits))
+ x)))
+ (+ n (log (scale-float (coerce f 'double-float)
+ (- sb!vm:double-float-digits))
+ 2.0d0))))))
+
(defun log (number &optional (base nil base-p))
#!+sb-doc
"Return the logarithm of NUMBER in the base BASE, which defaults to e."
(if base-p
- (if (zerop base)
- base ; ANSI spec
- (/ (log number) (log base)))
+ (cond
+ ((zerop base) 0f0) ; FIXME: type
+ ((and (typep number '(integer (0) *))
+ (typep base '(integer (0) *)))
+ (coerce (/ (log2 number) (log2 base)) 'single-float))
+ (t (/ (log number) (log base))))
(number-dispatch ((number number))
- (((foreach fixnum bignum ratio))
+ (((foreach fixnum bignum))
+ (if (minusp number)
+ (complex (log (- number)) (coerce pi 'single-float))
+ (coerce (/ (log2 number) (log (exp 1.0d0) 2.0d0)) 'single-float)))
+ ((ratio)
(if (minusp number)
(complex (log (- number)) (coerce pi 'single-float))
- (coerce (%log (coerce number 'double-float)) 'single-float)))
+ (let ((numerator (numerator number))
+ (denominator (denominator number)))
+ (if (= (integer-length numerator)
+ (integer-length denominator))
+ (coerce (%log1p (coerce (- number 1) 'double-float))
+ 'single-float)
+ (coerce (/ (- (log2 numerator) (log2 denominator))
+ (log (exp 1.0d0) 2.0d0))
+ 'single-float)))))
(((foreach single-float double-float))
;; Is (log -0) -infinity (libm.a) or -infinity + i*pi (Kahan)?
;; Since this doesn't seem to be an implementation issue
(coerce (%sqrt (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (minusp number)
- (complex-sqrt number)
+ (complex-sqrt (complex number))
(coerce (%sqrt (coerce number 'double-float))
'(dispatch-type number))))
((complex)
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
- (complex-asin number)
+ (complex-asin (complex number))
(coerce (%asin (coerce number 'double-float))
'(dispatch-type number))))
((complex)
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
- (complex-acos number)
+ (complex-acos (complex number))
(coerce (%acos (coerce number 'double-float))
'(dispatch-type number))))
((complex)
(float-sign y pi))
(float-sign y (/ pi 2)))
(%atan2 y x))))
- (number-dispatch ((y number) (x number))
+ (number-dispatch ((y real) (x real))
((double-float
(foreach double-float single-float fixnum bignum ratio))
(atan2 y (coerce x 'double-float)))
((complex)
(complex-atan y)))))
-;; It seems that everyone has a C version of sinh, cosh, and
-;; tanh. Let's use these for reals because the original
-;; implementations based on the definitions lose big in round-off
-;; error. These bad definitions also mean that sin and cos for
-;; complex numbers can also lose big.
+;;; It seems that every target system has a C version of sinh, cosh,
+;;; and tanh. Let's use these for reals because the original
+;;; implementations based on the definitions lose big in round-off
+;;; error. These bad definitions also mean that sin and cos for
+;;; complex numbers can also lose big.
(defun sinh (number)
#!+sb-doc
(coerce (%acosh (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (< number (coerce 1 '(dispatch-type number)))
- (complex-acosh number)
+ (complex-acosh (complex number))
(coerce (%acosh (coerce number 'double-float))
'(dispatch-type number))))
((complex)
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
(< number (coerce -1 '(dispatch-type number))))
- (complex-atanh number)
+ (complex-atanh (complex number))
(coerce (%atanh (coerce number 'double-float))
'(dispatch-type number))))
((complex)
;;;;
;;;; The original CMU CL code requested:
;;;; Please send any bug reports, comments, or improvements to
-;;;; Raymond Toy at toy@rtp.ericsson.se.
+;;;; Raymond Toy at <email address deleted during 2002 spam avalanche>.
;;; FIXME: In SBCL, the floating point infinity constants like
;;; SB!EXT:DOUBLE-FLOAT-POSITIVE-INFINITY aren't available as
;;; they're effectively implemented as special variable references,
;;; and the code below which uses them might be unnecessarily
;;; inefficient. Perhaps some sort of MAKE-LOAD-TIME-VALUE hackery
-;;; should be used instead?
+;;; should be used instead? (KLUDGED 2004-03-08 CSR, by replacing the
+;;; special variable references with (probably equally slow)
+;;; constructors)
+;;;
+;;; FIXME: As of 2004-05, when PFD noted that IMAGPART and COMPLEX
+;;; differ in their interpretations of the real line, IMAGPART was
+;;; patch, which without a certain amount of effort would have altered
+;;; all the branch cut treatment. Clients of these COMPLEX- routines
+;;; were patched to use explicit COMPLEX, rather than implicitly
+;;; passing in real numbers for treatment with IMAGPART, and these
+;;; COMPLEX- functions altered to require arguments of type COMPLEX;
+;;; however, someone needs to go back to Kahan for the definitive
+;;; answer for treatment of negative real floating point numbers and
+;;; branch cuts. If adjustment is needed, it is probably the removal
+;;; of explicit calls to COMPLEX in the clients of irrational
+;;; functions. -- a slightly bitter CSR, 2004-05-16
(declaim (inline square))
(defun square (x)
(cond ((float-nan-p x)
x)
((float-infinity-p x)
- sb!ext:double-float-positive-infinity)
+ ;; DOUBLE-FLOAT-POSITIVE-INFINITY
+ (double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0))
((zerop x)
;; The answer is negative infinity, but we are supposed to
;; signal divide-by-zero, so do the actual division
(defun coerce-to-complex-type (x y z)
(declare (double-float x y)
(number z))
- (if (subtypep (type-of (realpart z)) 'double-float)
+ (if (typep (realpart z) 'double-float)
(complex x y)
- ;; Convert anything that's not a DOUBLE-FLOAT to a SINGLE-FLOAT.
+ ;; Convert anything that's not already a DOUBLE-FLOAT (because
+ ;; the initial argument was a (COMPLEX DOUBLE-FLOAT) and we
+ ;; haven't done anything to lose precision) to a SINGLE-FLOAT.
(complex (float x 1f0)
(float y 1f0))))
(float-infinity-p rho))
(or (float-infinity-p (abs x))
(float-infinity-p (abs y))))
- (values sb!ext:double-float-positive-infinity 0))
+ ;; DOUBLE-FLOAT-POSITIVE-INFINITY
+ (values
+ (double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0)
+ 0))
((let ((threshold #.(/ least-positive-double-float
double-float-epsilon))
(traps (ldb sb!vm::float-sticky-bits
;;; principal square root of Z
;;;
-;;; Z may be any NUMBER, but the result is always a COMPLEX.
+;;; Z may be RATIONAL or COMPLEX; the result is always a COMPLEX.
(defun complex-sqrt (z)
- (declare (number z))
+ ;; KLUDGE: Here and below, we can't just declare Z to be of type
+ ;; COMPLEX, because one-arg COMPLEX on rationals returns a rational.
+ ;; Since there isn't a rational negative zero, this is OK from the
+ ;; point of view of getting the right answer in the face of branch
+ ;; cuts, but declarations of the form (OR RATIONAL COMPLEX) are
+ ;; still ugly. -- CSR, 2004-05-16
+ (declare (type (or complex rational) z))
(multiple-value-bind (rho k)
(cssqs z)
(declare (type (or (member 0d0) (double-float 0d0)) rho)
;;;
;;; This is for use with J /= 0 only when |z| is huge.
(defun complex-log-scaled (z j)
- (declare (number z)
+ (declare (type (or rational complex) z)
(fixnum j))
;; The constants t0, t1, t2 should be evaluated to machine
;; precision. In addition, Kahan says the accuracy of log1p
;;;
;;; Z may be any number, but the result is always a complex.
(defun complex-log (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
(complex-log-scaled z 0))
;;; KLUDGE: Let us note the following "strange" behavior. atanh 1.0d0
;;; i*y is never 0 since we have positive and negative zeroes. -- rtoy
;;; Compute atanh z = (log(1+z) - log(1-z))/2.
(defun complex-atanh (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
(let* (;; constants
(theta (/ (sqrt most-positive-double-float) 4.0d0))
(rho (/ 4.0d0 (sqrt most-positive-double-float)))
(declare (optimize (speed 3)))
(cond ((or (> x theta)
(> (abs y) theta))
- ;; To avoid overflow...
- (setf eta (float-sign y half-pi))
- ;; nu is real part of 1/(x + iy). This is x/(x^2+y^2),
+ ;; To avoid overflow...
+ (setf nu (float-sign y half-pi))
+ ;; ETA is real part of 1/(x + iy). This is x/(x^2+y^2),
;; which can cause overflow. Arrange this computation so
;; that it won't overflow.
- (setf nu (let* ((x-bigger (> x (abs y)))
- (r (if x-bigger (/ y x) (/ x y)))
- (d (+ 1.0d0 (* r r))))
- (if x-bigger
- (/ (/ x) d)
- (/ (/ r y) d)))))
+ (setf eta (let* ((x-bigger (> x (abs y)))
+ (r (if x-bigger (/ y x) (/ x y)))
+ (d (+ 1.0d0 (* r r))))
+ (if x-bigger
+ (/ (/ x) d)
+ (/ (/ r y) d)))))
((= x 1.0d0)
;; Should this be changed so that if y is zero, eta is set
;; to +infinity instead of approx 176? In any case
;; tanh(176) is 1.0d0 within working precision.
(let ((t1 (+ 4d0 (square y)))
(t2 (+ (abs y) rho)))
- (setf eta (log (/ (sqrt (sqrt t1)))
- (sqrt t2)))
+ (setf eta (log (/ (sqrt (sqrt t1))
+ (sqrt t2))))
(setf nu (* 0.5d0
(float-sign y
(+ half-pi (atan (* 0.5d0 t2))))))))
;;; Compute tanh z = sinh z / cosh z.
(defun complex-tanh (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
(let ((x (float (realpart z) 1.0d0))
(y (float (imagpart z) 1.0d0)))
(locally
;; space 0 to get maybe-inline functions inlined
(declare (optimize (speed 3) (space 0)))
(cond ((> (abs x)
- #-(or linux hpux) #.(/ (asinh most-positive-double-float) 4d0)
- ;; This is more accurate under linux.
- #+(or linux hpux) #.(/ (+ (log 2.0d0)
- (log most-positive-double-float)) 4d0))
- (coerce-to-complex-type (float-sign x)
- (float-sign y) z))
+ ;; FIXME: this form is hideously broken wrt
+ ;; cross-compilation portability. Much else in this
+ ;; file is too, of course, sometimes hidden by
+ ;; constant-folding, but this one in particular clearly
+ ;; depends on host and target
+ ;; MOST-POSITIVE-DOUBLE-FLOATs being equal. -- CSR,
+ ;; 2003-04-20
+ #.(/ (+ (log 2.0d0)
+ (log most-positive-double-float))
+ 4d0))
+ (coerce-to-complex-type (float-sign x)
+ (float-sign y) z))
(t
(let* ((tv (%tan y))
(beta (+ 1.0d0 (* tv tv)))
;;
;; and these two expressions are equal if and only if arg conj z =
;; -arg z, which is clearly true for all z.
- (declare (number z))
+ (declare (type (or rational complex) z))
(let ((sqrt-1+z (complex-sqrt (+ 1 z)))
(sqrt-1-z (complex-sqrt (- 1 z))))
(with-float-traps-masked (:divide-by-zero)
;;;
;;; Z may be any NUMBER, but the result is always a COMPLEX.
(defun complex-acosh (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
(let ((sqrt-z-1 (complex-sqrt (- z 1)))
(sqrt-z+1 (complex-sqrt (+ z 1))))
(with-float-traps-masked (:divide-by-zero)
;;;
;;; Z may be any NUMBER, but the result is always a COMPLEX.
(defun complex-asin (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
(let ((sqrt-1-z (complex-sqrt (- 1 z)))
(sqrt-1+z (complex-sqrt (+ 1 z))))
(with-float-traps-masked (:divide-by-zero)
;;;
;;; Z may be any number, but the result is always a complex.
(defun complex-asinh (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
;; asinh z = -i * asin (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
(result (complex-asin iz)))
;;;
;;; Z may be any number, but the result is always a complex.
(defun complex-atan (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
;; atan z = -i * atanh (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
(result (complex-atanh iz)))
;;;
;;; Z may be any number, but the result is always a complex.
(defun complex-tan (z)
- (declare (number z))
+ (declare (type (or rational complex) z))
;; tan z = -i * tanh(i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
(result (complex-tanh iz)))
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