\f
#!+x86 ;; for constant folding
(macrolet ((def (name ll)
- `(defun ,name ,ll (,name ,@ll))))
+ `(defun ,name ,ll (,name ,@ll))))
(def %atan2 (x y))
(def %atan (x))
(def %tan-quick (x))
(handle-reals %exp number)
((complex)
(* (exp (realpart number))
- (cis (imagpart number))))))
+ (cis (imagpart number))))))
;;; INTEXP -- Handle the rational base, integer power case.
;;; integers, and inverted if negative.
(defun intexp (base power)
(when (and *intexp-maximum-exponent*
- (> (abs power) *intexp-maximum-exponent*))
+ (> (abs power) *intexp-maximum-exponent*))
(error "The absolute value of ~S exceeds ~S."
- power '*intexp-maximum-exponent*))
+ power '*intexp-maximum-exponent*))
(cond ((minusp power)
- (/ (intexp base (- power))))
- ((eql base 2)
- (ash 1 power))
- (t
- (do ((nextn (ash power -1) (ash power -1))
- (total (if (oddp power) base 1)
- (if (oddp power) (* base total) total)))
- ((zerop nextn) total)
- (setq base (* base base))
- (setq power nextn)))))
+ (/ (intexp base (- power))))
+ ((eql base 2)
+ (ash 1 power))
+ (t
+ (do ((nextn (ash power -1) (ash power -1))
+ (total (if (oddp power) base 1)
+ (if (oddp power) (* base total) total)))
+ ((zerop nextn) total)
+ (setq base (* base base))
+ (setq power nextn)))))
;;; If an integer power of a rational, use INTEXP above. Otherwise, do
;;; floating point stuff. If both args are real, we try %POW right
"Return BASE raised to the POWER."
(if (zerop power)
(let ((result (1+ (* base power))))
- (if (and (floatp result) (float-nan-p result))
- (float 1 result)
- result))
+ (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
- ;; 2 - an even int
- (isint (ihi lo)
- (declare (type (unsigned-byte 31) ihi)
- (type (unsigned-byte 32) lo)
- (optimize (speed 3) (safety 0)))
- (let ((isint 0))
- (declare (type fixnum isint))
- (cond ((>= ihi #x43400000) ; exponent >= 53
- (setq isint 2))
- ((>= ihi #x3ff00000)
- (let ((k (- (ash ihi -20) #x3ff))) ; exponent
- (declare (type (mod 53) k))
- (cond ((> k 20)
- (let* ((shift (- 52 k))
- (j (logand (ash lo (- shift))))
- (j2 (ash j shift)))
- (declare (type (mod 32) shift)
- (type (unsigned-byte 32) j j2))
- (when (= j2 lo)
- (setq isint (- 2 (logand j 1))))))
- ((= lo 0)
- (let* ((shift (- 20 k))
- (j (ash ihi (- shift)))
- (j2 (ash j shift)))
- (declare (type (mod 32) shift)
- (type (unsigned-byte 31) j j2))
- (when (= j2 ihi)
- (setq isint (- 2 (logand j 1))))))))))
- isint))
- (real-expt (x y rtype)
- (let ((x (coerce x 'double-float))
- (y (coerce y 'double-float)))
- (declare (double-float x y))
- (let* ((x-hi (sb!kernel:double-float-high-bits x))
- (x-lo (sb!kernel:double-float-low-bits x))
- (x-ihi (logand x-hi #x7fffffff))
- (y-hi (sb!kernel:double-float-high-bits y))
- (y-lo (sb!kernel:double-float-low-bits y))
- (y-ihi (logand y-hi #x7fffffff)))
- (declare (type (signed-byte 32) x-hi y-hi)
- (type (unsigned-byte 31) x-ihi y-ihi)
- (type (unsigned-byte 32) x-lo y-lo))
- ;; y==zero: x**0 = 1
- (when (zerop (logior y-ihi y-lo))
- (return-from real-expt (coerce 1d0 rtype)))
- ;; +-NaN return x+y
- (when (or (> x-ihi #x7ff00000)
- (and (= x-ihi #x7ff00000) (/= x-lo 0))
- (> y-ihi #x7ff00000)
- (and (= y-ihi #x7ff00000) (/= y-lo 0)))
- (return-from real-expt (coerce (+ x y) rtype)))
- (let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
- (declare (type fixnum yisint))
- ;; special value of y
- (when (and (zerop y-lo) (= y-ihi #x7ff00000))
- ;; y is +-inf
- (return-from real-expt
- (cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
- ;; +-1**inf is NaN
- (coerce (- y y) rtype))
- ((>= x-ihi #x3ff00000)
- ;; (|x|>1)**+-inf = inf,0
- (if (>= y-hi 0)
- (coerce y rtype)
- (coerce 0 rtype)))
- (t
- ;; (|x|<1)**-,+inf = inf,0
- (if (< y-hi 0)
- (coerce (- y) rtype)
- (coerce 0 rtype))))))
+ ;; 0 - not an integer
+ ;; 1 - an odd int
+ ;; 2 - an even int
+ (isint (ihi lo)
+ (declare (type (unsigned-byte 31) ihi)
+ (type (unsigned-byte 32) lo)
+ (optimize (speed 3) (safety 0)))
+ (let ((isint 0))
+ (declare (type fixnum isint))
+ (cond ((>= ihi #x43400000) ; exponent >= 53
+ (setq isint 2))
+ ((>= ihi #x3ff00000)
+ (let ((k (- (ash ihi -20) #x3ff))) ; exponent
+ (declare (type (mod 53) k))
+ (cond ((> k 20)
+ (let* ((shift (- 52 k))
+ (j (logand (ash lo (- shift))))
+ (j2 (ash j shift)))
+ (declare (type (mod 32) shift)
+ (type (unsigned-byte 32) j j2))
+ (when (= j2 lo)
+ (setq isint (- 2 (logand j 1))))))
+ ((= lo 0)
+ (let* ((shift (- 20 k))
+ (j (ash ihi (- shift)))
+ (j2 (ash j shift)))
+ (declare (type (mod 32) shift)
+ (type (unsigned-byte 31) j j2))
+ (when (= j2 ihi)
+ (setq isint (- 2 (logand j 1))))))))))
+ isint))
+ (real-expt (x y rtype)
+ (let ((x (coerce x 'double-float))
+ (y (coerce y 'double-float)))
+ (declare (double-float x y))
+ (let* ((x-hi (sb!kernel:double-float-high-bits x))
+ (x-lo (sb!kernel:double-float-low-bits x))
+ (x-ihi (logand x-hi #x7fffffff))
+ (y-hi (sb!kernel:double-float-high-bits y))
+ (y-lo (sb!kernel:double-float-low-bits y))
+ (y-ihi (logand y-hi #x7fffffff)))
+ (declare (type (signed-byte 32) x-hi y-hi)
+ (type (unsigned-byte 31) x-ihi y-ihi)
+ (type (unsigned-byte 32) x-lo y-lo))
+ ;; y==zero: x**0 = 1
+ (when (zerop (logior y-ihi y-lo))
+ (return-from real-expt (coerce 1d0 rtype)))
+ ;; +-NaN return x+y
+ (when (or (> x-ihi #x7ff00000)
+ (and (= x-ihi #x7ff00000) (/= x-lo 0))
+ (> y-ihi #x7ff00000)
+ (and (= y-ihi #x7ff00000) (/= y-lo 0)))
+ (return-from real-expt (coerce (+ x y) rtype)))
+ (let ((yisint (if (< x-hi 0) (isint y-ihi y-lo) 0)))
+ (declare (type fixnum yisint))
+ ;; special value of y
+ (when (and (zerop y-lo) (= y-ihi #x7ff00000))
+ ;; y is +-inf
+ (return-from real-expt
+ (cond ((and (= x-ihi #x3ff00000) (zerop x-lo))
+ ;; +-1**inf is NaN
+ (coerce (- y y) rtype))
+ ((>= x-ihi #x3ff00000)
+ ;; (|x|>1)**+-inf = inf,0
+ (if (>= y-hi 0)
+ (coerce y rtype)
+ (coerce 0 rtype)))
+ (t
+ ;; (|x|<1)**-,+inf = inf,0
+ (if (< y-hi 0)
+ (coerce (- y) rtype)
+ (coerce 0 rtype))))))
- (let ((abs-x (abs x)))
- (declare (double-float abs-x))
- ;; special value of x
- (when (and (zerop x-lo)
- (or (= x-ihi #x7ff00000) (zerop x-ihi)
- (= x-ihi #x3ff00000)))
- ;; x is +-0,+-inf,+-1
- (let ((z (if (< y-hi 0)
- (/ 1 abs-x) ; z = (1/|x|)
- abs-x)))
- (declare (double-float z))
- (when (< x-hi 0)
- (cond ((and (= x-ihi #x3ff00000) (zerop yisint))
- ;; (-1)**non-int
- (let ((y*pi (* y pi)))
- (declare (double-float y*pi))
- (return-from real-expt
- (complex
- (coerce (%cos y*pi) rtype)
- (coerce (%sin y*pi) rtype)))))
- ((= yisint 1)
- ;; (x<0)**odd = -(|x|**odd)
- (setq z (- z)))))
- (return-from real-expt (coerce z rtype))))
+ (let ((abs-x (abs x)))
+ (declare (double-float abs-x))
+ ;; special value of x
+ (when (and (zerop x-lo)
+ (or (= x-ihi #x7ff00000) (zerop x-ihi)
+ (= x-ihi #x3ff00000)))
+ ;; x is +-0,+-inf,+-1
+ (let ((z (if (< y-hi 0)
+ (/ 1 abs-x) ; z = (1/|x|)
+ abs-x)))
+ (declare (double-float z))
+ (when (< x-hi 0)
+ (cond ((and (= x-ihi #x3ff00000) (zerop yisint))
+ ;; (-1)**non-int
+ (let ((y*pi (* y pi)))
+ (declare (double-float y*pi))
+ (return-from real-expt
+ (complex
+ (coerce (%cos y*pi) rtype)
+ (coerce (%sin y*pi) rtype)))))
+ ((= yisint 1)
+ ;; (x<0)**odd = -(|x|**odd)
+ (setq z (- z)))))
+ (return-from real-expt (coerce z rtype))))
- (if (>= x-hi 0)
- ;; x>0
- (coerce (sb!kernel::%pow x y) rtype)
- ;; x<0
- (let ((pow (sb!kernel::%pow abs-x y)))
- (declare (double-float pow))
- (case yisint
- (1 ; odd
- (coerce (* -1d0 pow) rtype))
- (2 ; even
- (coerce pow rtype))
- (t ; non-integer
- (let ((y*pi (* y pi)))
- (declare (double-float y*pi))
- (complex
- (coerce (* pow (%cos y*pi))
- rtype)
- (coerce (* pow (%sin y*pi))
- rtype)))))))))))))
+ (if (>= x-hi 0)
+ ;; x>0
+ (coerce (sb!kernel::%pow x y) rtype)
+ ;; x<0
+ (let ((pow (sb!kernel::%pow abs-x y)))
+ (declare (double-float pow))
+ (case yisint
+ (1 ; odd
+ (coerce (* -1d0 pow) rtype))
+ (2 ; even
+ (coerce pow rtype))
+ (t ; non-integer
+ (let ((y*pi (* y pi)))
+ (declare (double-float y*pi))
+ (complex
+ (coerce (* pow (%cos y*pi))
+ rtype)
+ (coerce (* pow (%sin y*pi))
+ rtype)))))))))))))
(declare (inline real-expt))
(number-dispatch ((base number) (power number))
- (((foreach fixnum (or bignum ratio) (complex rational)) integer)
- (intexp base power))
- (((foreach single-float double-float) rational)
- (real-expt base power '(dispatch-type base)))
- (((foreach fixnum (or bignum ratio) single-float)
- (foreach ratio single-float))
- (real-expt base power 'single-float))
- (((foreach fixnum (or bignum ratio) single-float double-float)
- double-float)
- (real-expt base power 'double-float))
- ((double-float single-float)
- (real-expt base power 'double-float))
- (((foreach (complex rational) (complex float)) rational)
- (* (expt (abs base) power)
- (cis (* power (phase base)))))
- (((foreach fixnum (or bignum ratio) single-float double-float)
- complex)
- (if (and (zerop base) (plusp (realpart power)))
- (* base power)
- (exp (* power (log base)))))
- (((foreach (complex float) (complex rational))
- (foreach complex double-float single-float))
- (if (and (zerop base) (plusp (realpart power)))
- (* base power)
- (exp (* power (log base)))))))))
+ (((foreach fixnum (or bignum ratio) (complex rational)) integer)
+ (intexp base power))
+ (((foreach single-float double-float) rational)
+ (real-expt base power '(dispatch-type base)))
+ (((foreach fixnum (or bignum ratio) single-float)
+ (foreach ratio single-float))
+ (real-expt base power 'single-float))
+ (((foreach fixnum (or bignum ratio) single-float double-float)
+ double-float)
+ (real-expt base power 'double-float))
+ ((double-float single-float)
+ (real-expt base power 'double-float))
+ (((foreach (complex rational) (complex float)) rational)
+ (* (expt (abs base) power)
+ (cis (* power (phase base)))))
+ (((foreach fixnum (or bignum ratio) single-float double-float)
+ complex)
+ (if (and (zerop base) (plusp (realpart power)))
+ (* base power)
+ (exp (* power (log base)))))
+ (((foreach (complex float) (complex rational))
+ (foreach complex double-float single-float))
+ (if (and (zerop base) (plusp (realpart power)))
+ (* base power)
+ (exp (* power (log base)))))))))
;;; FIXME: Maybe rename this so that it's clearer that it only works
;;; on integers?
;; 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))))))
+ (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
(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))))
+ ((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))
- (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))
- (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
- ;; I (pw) take the Kahan result.
- (if (< (float-sign number)
- (coerce 0 '(dispatch-type number)))
- (complex (log (- number)) (coerce pi '(dispatch-type number)))
- (coerce (%log (coerce number 'double-float))
- '(dispatch-type number))))
- ((complex)
- (complex-log number)))))
+ (((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))
+ (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
+ ;; I (pw) take the Kahan result.
+ (if (< (float-sign number)
+ (coerce 0 '(dispatch-type number)))
+ (complex (log (- number)) (coerce pi '(dispatch-type number)))
+ (coerce (%log (coerce number 'double-float))
+ '(dispatch-type number))))
+ ((complex)
+ (complex-log number)))))
(defun sqrt (number)
#!+sb-doc
(number-dispatch ((number number))
(((foreach fixnum bignum ratio))
(if (minusp number)
- (complex-sqrt number)
- (coerce (%sqrt (coerce number 'double-float)) 'single-float)))
+ (complex-sqrt number)
+ (coerce (%sqrt (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (minusp number)
- (complex-sqrt (complex number))
- (coerce (%sqrt (coerce number 'double-float))
- '(dispatch-type number))))
+ (complex-sqrt (complex number))
+ (coerce (%sqrt (coerce number 'double-float))
+ '(dispatch-type number))))
((complex)
(complex-sqrt number))))
\f
(abs number))
((complex)
(let ((rx (realpart number))
- (ix (imagpart number)))
+ (ix (imagpart number)))
(etypecase rx
- (rational
- (sqrt (+ (* rx rx) (* ix ix))))
- (single-float
- (coerce (%hypot (coerce rx 'double-float)
- (coerce ix 'double-float))
- 'single-float))
- (double-float
- (%hypot rx ix)))))))
+ (rational
+ (sqrt (+ (* rx rx) (* ix ix))))
+ (single-float
+ (coerce (%hypot (coerce rx 'double-float)
+ (coerce ix 'double-float))
+ 'single-float))
+ (double-float
+ (%hypot rx ix)))))))
(defun phase (number)
#!+sb-doc
(etypecase number
(rational
(if (minusp number)
- (coerce pi 'single-float)
- 0.0f0))
+ (coerce pi 'single-float)
+ 0.0f0))
(single-float
(if (minusp (float-sign number))
- (coerce pi 'single-float)
- 0.0f0))
+ (coerce pi 'single-float)
+ 0.0f0))
(double-float
(if (minusp (float-sign number))
- (coerce pi 'double-float)
- 0.0d0))
+ (coerce pi 'double-float)
+ 0.0d0))
(complex
(atan (imagpart number) (realpart number)))))
(handle-reals %sin number)
((complex)
(let ((x (realpart number))
- (y (imagpart number)))
+ (y (imagpart number)))
(complex (* (sin x) (cosh y))
- (* (cos x) (sinh y)))))))
+ (* (cos x) (sinh y)))))))
(defun cos (number)
#!+sb-doc
(handle-reals %cos number)
((complex)
(let ((x (realpart number))
- (y (imagpart number)))
+ (y (imagpart number)))
(complex (* (cos x) (cosh y))
- (- (* (sin x) (sinh y))))))))
+ (- (* (sin x) (sinh y))))))))
(defun tan (number)
#!+sb-doc
(number-dispatch ((number number))
((rational)
(if (or (> number 1) (< number -1))
- (complex-asin number)
- (coerce (%asin (coerce number 'double-float)) 'single-float)))
+ (complex-asin number)
+ (coerce (%asin (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
- (< number (coerce -1 '(dispatch-type number))))
- (complex-asin (complex number))
- (coerce (%asin (coerce number 'double-float))
- '(dispatch-type number))))
+ (< number (coerce -1 '(dispatch-type number))))
+ (complex-asin (complex number))
+ (coerce (%asin (coerce number 'double-float))
+ '(dispatch-type number))))
((complex)
(complex-asin number))))
(number-dispatch ((number number))
((rational)
(if (or (> number 1) (< number -1))
- (complex-acos number)
- (coerce (%acos (coerce number 'double-float)) 'single-float)))
+ (complex-acos number)
+ (coerce (%acos (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
- (< number (coerce -1 '(dispatch-type number))))
- (complex-acos (complex number))
- (coerce (%acos (coerce number 'double-float))
- '(dispatch-type number))))
+ (< number (coerce -1 '(dispatch-type number))))
+ (complex-acos (complex number))
+ (coerce (%acos (coerce number 'double-float))
+ '(dispatch-type number))))
((complex)
(complex-acos number))))
"Return the arc tangent of Y if X is omitted or Y/X if X is supplied."
(if xp
(flet ((atan2 (y x)
- (declare (type double-float y x)
- (values double-float))
- (if (zerop x)
- (if (zerop y)
- (if (plusp (float-sign x))
- y
- (float-sign y pi))
- (float-sign y (/ pi 2)))
- (%atan2 y x))))
- (number-dispatch ((y real) (x real))
- ((double-float
- (foreach double-float single-float fixnum bignum ratio))
- (atan2 y (coerce x 'double-float)))
- (((foreach single-float fixnum bignum ratio)
- double-float)
- (atan2 (coerce y 'double-float) x))
- (((foreach single-float fixnum bignum ratio)
- (foreach single-float fixnum bignum ratio))
- (coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
- 'single-float))))
+ (declare (type double-float y x)
+ (values double-float))
+ (if (zerop x)
+ (if (zerop y)
+ (if (plusp (float-sign x))
+ y
+ (float-sign y pi))
+ (float-sign y (/ pi 2)))
+ (%atan2 y x))))
+ (number-dispatch ((y real) (x real))
+ ((double-float
+ (foreach double-float single-float fixnum bignum ratio))
+ (atan2 y (coerce x 'double-float)))
+ (((foreach single-float fixnum bignum ratio)
+ double-float)
+ (atan2 (coerce y 'double-float) x))
+ (((foreach single-float fixnum bignum ratio)
+ (foreach single-float fixnum bignum ratio))
+ (coerce (atan2 (coerce y 'double-float) (coerce x 'double-float))
+ 'single-float))))
(number-dispatch ((y number))
- (handle-reals %atan y)
- ((complex)
- (complex-atan y)))))
+ (handle-reals %atan y)
+ ((complex)
+ (complex-atan y)))))
;;; It seems that every target system has a C version of sinh, cosh,
;;; and tanh. Let's use these for reals because the original
(handle-reals %sinh number)
((complex)
(let ((x (realpart number))
- (y (imagpart number)))
+ (y (imagpart number)))
(complex (* (sinh x) (cos y))
- (* (cosh x) (sin y)))))))
+ (* (cosh x) (sin y)))))))
(defun cosh (number)
#!+sb-doc
(handle-reals %cosh number)
((complex)
(let ((x (realpart number))
- (y (imagpart number)))
+ (y (imagpart number)))
(complex (* (cosh x) (cos y))
- (* (sinh x) (sin y)))))))
+ (* (sinh x) (sin y)))))))
(defun tanh (number)
#!+sb-doc
((rational)
;; acosh is complex if number < 1
(if (< number 1)
- (complex-acosh number)
- (coerce (%acosh (coerce number 'double-float)) 'single-float)))
+ (complex-acosh number)
+ (coerce (%acosh (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (< number (coerce 1 '(dispatch-type number)))
- (complex-acosh (complex number))
- (coerce (%acosh (coerce number 'double-float))
- '(dispatch-type number))))
+ (complex-acosh (complex number))
+ (coerce (%acosh (coerce number 'double-float))
+ '(dispatch-type number))))
((complex)
(complex-acosh number))))
((rational)
;; atanh is complex if |number| > 1
(if (or (> number 1) (< number -1))
- (complex-atanh number)
- (coerce (%atanh (coerce number 'double-float)) 'single-float)))
+ (complex-atanh number)
+ (coerce (%atanh (coerce number 'double-float)) 'single-float)))
(((foreach single-float double-float))
(if (or (> number (coerce 1 '(dispatch-type number)))
- (< number (coerce -1 '(dispatch-type number))))
- (complex-atanh (complex number))
- (coerce (%atanh (coerce number 'double-float))
- '(dispatch-type number))))
+ (< number (coerce -1 '(dispatch-type number))))
+ (complex-atanh (complex number))
+ (coerce (%atanh (coerce number 'double-float))
+ '(dispatch-type number))))
((complex)
(complex-atanh number))))
;;; HP-UX does not supply a C version of log1p, so use the definition.
-;;;
+;;;
;;; FIXME: This is really not a good definition. As per Raymond Toy
;;; working on CMU CL, "The definition really loses big-time in
;;; roundoff as x gets small."
#!+hpux
(defun %log1p (number)
(declare (double-float number)
- (optimize (speed 3) (safety 0)))
+ (optimize (speed 3) (safety 0)))
(the double-float (log (the (double-float 0d0) (+ number 1d0)))))
\f
;;;; not-OLD-SPECFUN stuff
(declaim (inline scalb))
(defun scalb (x n)
(declare (type double-float x)
- (type double-float-exponent n))
+ (type double-float-exponent n))
(scale-float x n))
;;; This is like LOGB, but X is not infinity and non-zero and not a
(defun logb (x)
(declare (type double-float x))
(cond ((float-nan-p x)
- x)
- ((float-infinity-p x)
- ;; 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
+ x)
+ ((float-infinity-p x)
+ ;; 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
- (/ -1.0d0 x)
- )
- (t
+ (/ -1.0d0 x)
+ )
+ (t
(logb-finite x))))
;;; This function is used to create a complex number of the
;;; such that has the same type as Z. If Z has type (complex
;;; rational), the X and Y are coerced to single-float.
#!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
- (error "needs work for long float support"))
+ (error "needs work for long float support"))
(declaim (inline coerce-to-complex-type))
(defun coerce-to-complex-type (x y z)
(declare (double-float x y)
- (number z))
+ (number z))
(if (typep (realpart z) 'double-float)
(complex x y)
;; 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 y 1f0))))
;;; Compute |(x+i*y)/2^k|^2 scaled to avoid over/underflow. The
;;; result is r + i*k, where k is an integer.
#!+long-float (eval-when (:compile-toplevel :load-toplevel :execute)
- (error "needs work for long float support"))
+ (error "needs work for long float support"))
(defun cssqs (z)
(let ((x (float (realpart z) 1d0))
- (y (float (imagpart z) 1d0)))
+ (y (float (imagpart z) 1d0)))
;; Would this be better handled using an exception handler to
;; catch the overflow or underflow signal? For now, we turn all
;; traps off and look at the accrued exceptions to see if any
(let ((rho (+ (square x) (square y))))
(declare (optimize (speed 3) (space 0)))
(cond ((and (or (float-nan-p rho)
- (float-infinity-p rho))
- (or (float-infinity-p (abs x))
- (float-infinity-p (abs y))))
- ;; 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
- (sb!vm:floating-point-modes))))
+ (float-infinity-p rho))
+ (or (float-infinity-p (abs x))
+ (float-infinity-p (abs y))))
+ ;; 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
+ (sb!vm:floating-point-modes))))
;; Overflow raised or (underflow raised and rho <
;; lambda/eps)
- (or (not (zerop (logand sb!vm:float-overflow-trap-bit traps)))
- (and (not (zerop (logand sb!vm:float-underflow-trap-bit
- traps)))
- (< rho threshold))))
+ (or (not (zerop (logand sb!vm:float-overflow-trap-bit traps)))
+ (and (not (zerop (logand sb!vm:float-underflow-trap-bit
+ traps)))
+ (< rho threshold))))
;; If we're here, neither x nor y are infinity and at
;; least one is non-zero.. Thus logb returns a nice
;; integer.
(declare (type (or (member 0d0) (double-float 0d0)) rho)
(type fixnum k))
(let ((x (float (realpart z) 1.0d0))
- (y (float (imagpart z) 1.0d0))
- (eta 0d0)
- (nu 0d0))
+ (y (float (imagpart z) 1.0d0))
+ (eta 0d0)
+ (nu 0d0))
(declare (double-float x y eta nu))
(locally
(declare (optimize (speed 3) (space 0)))
(if (not (float-nan-p x))
- (setf rho (+ (scalb (abs x) (- k)) (sqrt rho))))
+ (setf rho (+ (scalb (abs x) (- k)) (sqrt rho))))
(cond ((oddp k)
- (setf k (ash k -1)))
- (t
- (setf k (1- (ash k -1)))
- (setf rho (+ rho rho))))
+ (setf k (ash k -1)))
+ (t
+ (setf k (1- (ash k -1)))
+ (setf rho (+ rho rho))))
(setf rho (scalb (sqrt rho) k))
(setf nu y)
(when (/= rho 0d0)
- (when (not (float-infinity-p (abs nu)))
- (setf nu (/ (/ nu rho) 2d0)))
- (when (< x 0d0)
- (setf eta (abs nu))
- (setf nu (float-sign y rho))))
+ (when (not (float-infinity-p (abs nu)))
+ (setf nu (/ (/ nu rho) 2d0)))
+ (when (< x 0d0)
+ (setf eta (abs nu))
+ (setf nu (float-sign y rho))))
(coerce-to-complex-type eta nu z)))))
-
+
;;; Compute log(2^j*z).
;;;
;;; This is for use with J /= 0 only when |z| is huge.
(defun complex-log-scaled (z j)
(declare (type (or rational complex) z)
- (fixnum j))
+ (fixnum j))
;; The constants t0, t1, t2 should be evaluated to machine
;; precision. In addition, Kahan says the accuracy of log1p
;; influences the choices of these constants but doesn't say how to
;; choose them. We'll just assume his choices matches our
;; implementation of log1p.
(let ((t0 #.(/ 1 (sqrt 2.0d0)))
- (t1 1.2d0)
- (t2 3d0)
- (ln2 #.(log 2d0))
- (x (float (realpart z) 1.0d0))
- (y (float (imagpart z) 1.0d0)))
+ (t1 1.2d0)
+ (t2 3d0)
+ (ln2 #.(log 2d0))
+ (x (float (realpart z) 1.0d0))
+ (y (float (imagpart z) 1.0d0)))
(multiple-value-bind (rho k)
- (cssqs z)
+ (cssqs z)
(declare (optimize (speed 3)))
(let ((beta (max (abs x) (abs y)))
- (theta (min (abs x) (abs y))))
+ (theta (min (abs x) (abs y))))
(coerce-to-complex-type (if (and (zerop k)
- (< t0 beta)
- (or (<= beta t1)
- (< rho t2)))
+ (< t0 beta)
+ (or (<= beta t1)
+ (< rho t2)))
(/ (%log1p (+ (* (- beta 1.0d0)
- (+ beta 1.0d0))
- (* theta theta)))
+ (+ beta 1.0d0))
+ (* theta theta)))
2d0)
(+ (/ (log rho) 2d0)
(* (+ k j) ln2)))
(defun complex-log (z)
(declare (type (or rational complex) z))
(complex-log-scaled z 0))
-
+
;;; KLUDGE: Let us note the following "strange" behavior. atanh 1.0d0
;;; is +infinity, but the following code returns approx 176 + i*pi/4.
;;; The reason for the imaginary part is caused by the fact that arg
(theta (/ (sqrt most-positive-double-float) 4.0d0))
(rho (/ 4.0d0 (sqrt most-positive-double-float)))
(half-pi (/ pi 2.0d0))
- (rp (float (realpart z) 1.0d0))
- (beta (float-sign rp 1.0d0))
- (x (* beta rp))
- (y (* beta (- (float (imagpart z) 1.0d0))))
- (eta 0.0d0)
- (nu 0.0d0))
+ (rp (float (realpart z) 1.0d0))
+ (beta (float-sign rp 1.0d0))
+ (x (* beta rp))
+ (y (* beta (- (float (imagpart z) 1.0d0))))
+ (eta 0.0d0)
+ (nu 0.0d0))
;; Shouldn't need this declare.
(declare (double-float x y))
(locally
(declare (optimize (speed 3)))
(cond ((or (> x theta)
- (> (abs y) theta))
- ;; 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 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 nu (* 0.5d0
- (float-sign y
- (+ half-pi (atan (* 0.5d0 t2))))))))
- (t
- (let ((t1 (+ (abs y) rho)))
+ (> (abs y) theta))
+ ;; 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 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 nu (* 0.5d0
+ (float-sign y
+ (+ half-pi (atan (* 0.5d0 t2))))))))
+ (t
+ (let ((t1 (+ (abs y) rho)))
;; Normal case using log1p(x) = log(1 + x)
- (setf eta (* 0.25d0
- (%log1p (/ (* 4.0d0 x)
- (+ (square (- 1.0d0 x))
- (square t1))))))
- (setf nu (* 0.5d0
- (atan (* 2.0d0 y)
- (- (* (- 1.0d0 x)
- (+ 1.0d0 x))
- (square t1))))))))
+ (setf eta (* 0.25d0
+ (%log1p (/ (* 4.0d0 x)
+ (+ (square (- 1.0d0 x))
+ (square t1))))))
+ (setf nu (* 0.5d0
+ (atan (* 2.0d0 y)
+ (- (* (- 1.0d0 x)
+ (+ 1.0d0 x))
+ (square t1))))))))
(coerce-to-complex-type (* beta eta)
- (- (* beta nu))
+ (- (* beta nu))
z))))
;;; Compute tanh z = sinh z / cosh z.
(defun complex-tanh (z)
(declare (type (or rational complex) z))
(let ((x (float (realpart z) 1.0d0))
- (y (float (imagpart 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)
- ;; 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)))
- (s (sinh x))
- (rho (sqrt (+ 1.0d0 (* s s)))))
- (if (float-infinity-p (abs tv))
- (coerce-to-complex-type (/ rho s)
- (/ tv)
- z)
- (let ((den (+ 1.0d0 (* beta s s))))
- (coerce-to-complex-type (/ (* beta rho s)
- den)
- (/ tv den)
+ ;; 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)))
+ (s (sinh x))
+ (rho (sqrt (+ 1.0d0 (* s s)))))
+ (if (float-infinity-p (abs tv))
+ (coerce-to-complex-type (/ rho s)
+ (/ tv)
+ z)
+ (let ((den (+ 1.0d0 (* beta s s))))
+ (coerce-to-complex-type (/ (* beta rho s)
+ den)
+ (/ tv den)
z)))))))))
;;; Compute acos z = pi/2 - asin z.
;; -arg z, which is clearly true for all z.
(declare (type (or rational complex) z))
(let ((sqrt-1+z (complex-sqrt (+ 1 z)))
- (sqrt-1-z (complex-sqrt (- 1 z))))
+ (sqrt-1-z (complex-sqrt (- 1 z))))
(with-float-traps-masked (:divide-by-zero)
(complex (* 2 (atan (/ (realpart sqrt-1-z)
- (realpart sqrt-1+z))))
- (asinh (imagpart (* (conjugate sqrt-1+z)
- sqrt-1-z)))))))
+ (realpart sqrt-1+z))))
+ (asinh (imagpart (* (conjugate sqrt-1+z)
+ sqrt-1-z)))))))
;;; Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
;;;
(defun complex-acosh (z)
(declare (type (or rational complex) z))
(let ((sqrt-z-1 (complex-sqrt (- z 1)))
- (sqrt-z+1 (complex-sqrt (+ z 1))))
+ (sqrt-z+1 (complex-sqrt (+ z 1))))
(with-float-traps-masked (:divide-by-zero)
(complex (asinh (realpart (* (conjugate sqrt-z-1)
- sqrt-z+1)))
- (* 2 (atan (/ (imagpart sqrt-z-1)
- (realpart sqrt-z+1))))))))
+ sqrt-z+1)))
+ (* 2 (atan (/ (imagpart sqrt-z-1)
+ (realpart sqrt-z+1))))))))
;;; Compute asin z = asinh(i*z)/i.
;;;
(defun complex-asin (z)
(declare (type (or rational complex) z))
(let ((sqrt-1-z (complex-sqrt (- 1 z)))
- (sqrt-1+z (complex-sqrt (+ 1 z))))
+ (sqrt-1+z (complex-sqrt (+ 1 z))))
(with-float-traps-masked (:divide-by-zero)
(complex (atan (/ (realpart z)
- (realpart (* sqrt-1-z sqrt-1+z))))
- (asinh (imagpart (* (conjugate sqrt-1-z)
- sqrt-1+z)))))))
+ (realpart (* sqrt-1-z sqrt-1+z))))
+ (asinh (imagpart (* (conjugate sqrt-1-z)
+ sqrt-1+z)))))))
;;; Compute asinh z = log(z + sqrt(1 + z*z)).
;;;
(declare (type (or rational complex) z))
;; asinh z = -i * asin (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
- (result (complex-asin iz)))
+ (result (complex-asin iz)))
(complex (imagpart result)
- (- (realpart result)))))
-
+ (- (realpart result)))))
+
;;; Compute atan z = atanh (i*z) / i.
;;;
;;; Z may be any number, but the result is always a complex.
(declare (type (or rational complex) z))
;; atan z = -i * atanh (i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
- (result (complex-atanh iz)))
+ (result (complex-atanh iz)))
(complex (imagpart result)
- (- (realpart result)))))
+ (- (realpart result)))))
;;; Compute tan z = -i * tanh(i * z)
;;;
(declare (type (or rational complex) z))
;; tan z = -i * tanh(i*z)
(let* ((iz (complex (- (imagpart z)) (realpart z)))
- (result (complex-tanh iz)))
+ (result (complex-tanh iz)))
(complex (imagpart result)
- (- (realpart result)))))
+ (- (realpart result)))))
projects / sbcl.git / blobdiff