\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.
(eval-when (:compile-toplevel :execute)
(sb!xc:defmacro def-math-rtn (name num-args)
- (let ((function (symbolicate "%" (string-upcase name))))
+ (let ((function (symbolicate "%" (string-upcase name)))
+ (args (loop for i below num-args
+ collect (intern (format nil "ARG~D" i)))))
`(progn
- (proclaim '(inline ,function))
- (sb!alien:def-alien-routine (,name ,function) double-float
- ,@(let ((results nil))
- (dotimes (i num-args (nreverse results))
- (push (list (intern (format nil "ARG-~D" i))
- 'double-float)
- results)))))))
+ (declaim (inline ,function))
+ (defun ,function ,args
+ (alien-funcall
+ (extern-alien ,name
+ (function double-float
+ ,@(loop repeat num-args
+ collect 'double-float)))
+ ,@args)))))
(defun handle-reals (function var)
`((((foreach fixnum single-float bignum ratio))
) ; 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 (x))
+ (def %tan-quick (x))
+ (def %cos (x))
+ (def %cos-quick (x))
+ (def %sin (x))
+ (def %sin-quick (x))
+ (def %sqrt (x))
+ (def %log (x))
+ (def %exp (x)))
+
+#!+x86-64 ;; for constant folding
+(macrolet ((def (name ll)
+ `(defun ,name ,ll (,name ,@ll))))
+ (def %sqrt (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)
#!-x86 (def-math-rtn "cos" 1)
#!-x86 (def-math-rtn "tan" 1)
-(def-math-rtn "asin" 1)
-(def-math-rtn "acos" 1)
#!-x86 (def-math-rtn "atan" 1)
#!-x86 (def-math-rtn "atan2" 2)
-(def-math-rtn "sinh" 1)
-(def-math-rtn "cosh" 1)
-(def-math-rtn "tanh" 1)
-(def-math-rtn "asinh" 1)
-(def-math-rtn "acosh" 1)
-(def-math-rtn "atanh" 1)
+#!-(and win32 x86)
+(progn
+ (def-math-rtn "acos" 1)
+ (def-math-rtn "asin" 1)
+ (def-math-rtn "cosh" 1)
+ (def-math-rtn "sinh" 1)
+ (def-math-rtn "tanh" 1)
+ #!-win32
+ (progn
+ (def-math-rtn "asinh" 1)
+ (def-math-rtn "acosh" 1)
+ (def-math-rtn "atanh" 1)))
+#!+win32
+(progn
+ #!-x86-64
+ (progn
+ (declaim (inline %asin))
+ (defun %asin (number)
+ (%atan (/ number (sqrt (- 1 (* number number))))))
+ (declaim (inline %acos))
+ (defun %acos (number)
+ (- (/ pi 2) (%asin number)))
+ (declaim (inline %cosh))
+ (defun %cosh (number)
+ (/ (+ (exp number) (exp (- number))) 2))
+ (declaim (inline %sinh))
+ (defun %sinh (number)
+ (/ (- (exp number) (exp (- number))) 2))
+ (declaim (inline %tanh))
+ (defun %tanh (number)
+ (/ (%sinh number) (%cosh number))))
+ (declaim (inline %asinh))
+ (defun %asinh (number)
+ (log (+ number (sqrt (+ (* number number) 1.0d0))) #.(exp 1.0d0)))
+ (declaim (inline %acosh))
+ (defun %acosh (number)
+ (log (+ number (sqrt (- (* number number) 1.0d0))) #.(exp 1.0d0)))
+ (declaim (inline %atanh))
+ (defun %atanh (number)
+ (let ((ratio (/ (+ 1 number) (- 1 number))))
+ ;; Were we effectively zero?
+ (if (= ratio -1.0d0)
+ 0.0d0
+ (/ (log ratio #.(exp 1.0d0)) 2.0d0)))))
;;; exponential and logarithmic
#!-x86 (def-math-rtn "exp" 1)
#!-x86 (def-math-rtn "log" 1)
#!-x86 (def-math-rtn "log10" 1)
-(def-math-rtn "pow" 2)
-#!-x86 (def-math-rtn "sqrt" 1)
-(def-math-rtn "hypot" 2)
-#!-(or hpux x86) (def-math-rtn "log1p" 1)
+#!-(and win32 x86) (def-math-rtn "pow" 2)
+#!-(or x86 x86-64) (def-math-rtn "sqrt" 1)
+#!-win32 (def-math-rtn "hypot" 2)
+#!-x86 (def-math-rtn "log1p" 1)
-#!+x86 ;; These are needed for use by byte-compiled files.
+#!+win32
(progn
- (defun %sin (x)
- (declare (double-float x)
- (values double-float))
- (%sin x))
- (defun %sin-quick (x)
- (declare (double-float x)
- (values double-float))
- (%sin-quick x))
- (defun %cos (x)
- (declare (double-float x)
- (values double-float))
- (%cos x))
- (defun %cos-quick (x)
- (declare (double-float x)
- (values double-float))
- (%cos-quick x))
- (defun %tan (x)
- (declare (double-float x)
- (values double-float))
- (%tan x))
- (defun %tan-quick (x)
- (declare (double-float x)
- (values double-float))
- (%tan-quick x))
- (defun %atan (x)
- (declare (double-float x)
- (values double-float))
- (%atan x))
- (defun %atan2 (x y)
- (declare (double-float x y)
- (values double-float))
- (%atan2 x y))
- (defun %exp (x)
- (declare (double-float x)
- (values double-float))
- (%exp x))
- (defun %log (x)
- (declare (double-float x)
- (values double-float))
- (%log x))
- (defun %log10 (x)
- (declare (double-float x)
- (values double-float))
- (%log10 x))
- #+nil ;; notyet
- (defun %pow (x y)
- (declare (type (double-float 0d0) x)
- (double-float y)
- (values (double-float 0d0)))
- (%pow x y))
- (defun %sqrt (x)
- (declare (double-float x)
- (values double-float))
- (%sqrt x))
- (defun %scalbn (f ex)
- (declare (double-float f)
- (type (signed-byte 32) ex)
- (values double-float))
- (%scalbn f ex))
- (defun %scalb (f ex)
- (declare (double-float f ex)
- (values double-float))
- (%scalb f ex))
- (defun %logb (x)
- (declare (double-float x)
- (values double-float))
- (%logb x))
- (defun %log1p (x)
- (declare (double-float x)
- (values double-float))
- (%log1p x))
- ) ; progn
+ ;; This is written in a peculiar way to avoid overflow. Note that in
+ ;; sqrt(x^2 + y^2), either square or the sum can overflow.
+ ;;
+ ;; Factoring x^2 out of sqrt(x^2 + y^2) gives us the expression
+ ;; |x|sqrt(1 + (y/x)^2), which, assuming |x| >= |y|, can only overflow
+ ;; if |x| is sufficiently large.
+ ;;
+ ;; The ZEROP test suffices (y is non-negative) to guard against
+ ;; divisions by zero: x >= y > 0.
+ (declaim (inline %hypot))
+ (defun %hypot (x y)
+ (declare (type double-float x y))
+ (let ((x (abs x))
+ (y (abs y)))
+ (when (> y x)
+ (rotatef x y))
+ (if (zerop y)
+ x
+ (let ((y/x (/ y x)))
+ (* x (sqrt (1+ (* y/x y/x)))))))))
\f
;;;; power functions
(handle-reals %exp number)
((complex)
(* (exp (realpart number))
- (cis (imagpart number))))))
+ (cis (imagpart number))))))
;;; 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)
- (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
;;; from the general complex case.
(defun expt (base power)
#!+sb-doc
- "Returns BASE raised to the POWER."
+ "Return BASE raised to the POWER."
(if (zerop power)
- (1+ (* base power))
+ (if (and (zerop base) (floatp power))
+ (error 'arguments-out-of-domain-error
+ :operands (list base power)
+ :operation 'expt
+ :references (list '(:ansi-cl :function expt)))
+ (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
- ;; 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))))
-
- (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))
+ ;; 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
+ ;; FIXME: Hardcoded qNaN/sNaN values are not portable.
+ (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))))
+
+ (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))))))))))))
+ (complex-expt (base power)
+ (if (and (zerop base) (plusp (realpart power)))
+ (* base power)
+ (exp (* power (log base))))))
+ (declare (inline real-expt complex-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))
+ ;; Handle (expt <complex> <rational>), except the case dealt with
+ ;; in the first clause above, (expt <(complex rational)> <integer>).
+ (((foreach (complex rational) (complex single-float)
+ (complex double-float))
+ rational)
+ (* (expt (abs base) power)
+ (cis (* power (phase base)))))
+ ;; The next three clauses handle (expt <real> <complex>).
+ (((foreach fixnum (or bignum ratio) single-float)
+ (foreach (complex single-float) (complex rational)))
+ (complex-expt base power))
+ (((foreach fixnum (or bignum ratio) single-float)
+ (complex double-float))
+ (complex-expt (coerce base 'double-float) power))
+ ((double-float complex)
+ (complex-expt base power))
+ ;; The next three clauses handle (expt <complex> <float>) and
+ ;; (expt <complex> <complex>).
+ (((foreach (complex single-float) (complex rational))
+ (foreach (complex single-float) (complex rational) single-float))
+ (complex-expt base power))
+ (((foreach (complex single-float) (complex rational))
+ (foreach (complex double-float) double-float))
+ (complex-expt (coerce base '(complex double-float)) power))
+ (((complex double-float)
+ (foreach complex double-float single-float))
+ (complex-expt base power))))))
+
+;;; 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)
+ (if (or (typep number 'double-float) (typep base 'double-float))
+ 0.0d0
+ 0.0f0))
+ ((and (typep number '(integer (0) *))
+ (typep base '(integer (0) *)))
+ (coerce (/ (log2 number) (log2 base)) 'single-float))
+ ((and (typep number 'integer) (typep base 'double-float))
+ ;; No single float intermediate result
+ (/ (log2 number) (log base 2.0d0)))
+ ((and (typep number 'double-float) (typep base 'integer))
+ (/ (log number 2.0d0) (log2 base)))
+ (t
+ (/ (log number) (log base))))
(number-dispatch ((number number))
- (((foreach fixnum bignum ratio))
- (if (minusp number)
- (complex (log (- number)) (coerce pi 'single-float))
- (coerce (%log (coerce number 'double-float)) '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 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
(defun abs (number)
#!+sb-doc
- "Returns the absolute value of the number."
+ "Return the absolute value of the number."
(number-dispatch ((number number))
(((foreach single-float double-float fixnum rational))
(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
(defun cis (theta)
#!+sb-doc
- "Return cos(Theta) + i sin(Theta), AKA exp(i Theta)."
+ "Return cos(Theta) + i sin(Theta), i.e. exp(i Theta)."
(declare (type real theta))
(complex (cos theta) (sin theta)))
(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 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 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 number) (x number))
- ((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 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.
-
-#+nil
-(defun sinh (number)
- #!+sb-doc
- "Return the hyperbolic sine of NUMBER."
- (/ (- (exp number) (exp (- number))) 2))
+;;; 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
(handle-reals %sinh number)
((complex)
(let ((x (realpart number))
- (y (imagpart number)))
+ (y (imagpart number)))
(complex (* (sinh x) (cos y))
- (* (cosh x) (sin y)))))))
-
-#+nil
-(defun cosh (number)
- #!+sb-doc
- "Return the hyperbolic cosine of NUMBER."
- (/ (+ (exp number) (exp (- number))) 2))
+ (* (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 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 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.
-#!+hpux
-#!-sb-fluid (declaim (inline %log1p))
-#!+hpux
-(defun %log1p (number)
- (declare (double-float number)
- (optimize (speed 3) (safety 0)))
- (the double-float (log (the (double-float 0d0) (+ number 1d0)))))
\f
-;;;; OLD-SPECFUN stuff
+;;;; not-OLD-SPECFUN stuff
;;;;
;;;; (This was conditional on #-OLD-SPECFUN in the CMU CL sources,
;;;; but OLD-SPECFUN was mentioned nowhere else, so it seems to be
;;;;
;;;; 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))
-(declaim (ftype (function (double-float) (double-float 0d0)) square))
(defun square (x)
- (declare (double-float x)
- (values (double-float 0d0)))
+ (declare (double-float x))
(* x x))
;;; original CMU CL comment, apparently re. SCALB and LOGB and
(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
+;;; NaN, so we can always return an integer.
+(declaim (inline logb-finite))
+(defun logb-finite (x)
+ (declare (type double-float x))
+ (multiple-value-bind (signif exponent sign)
+ (decode-float x)
+ (declare (ignore signif sign))
+ ;; DECODE-FLOAT is almost right, except that the exponent is off
+ ;; by one.
+ (1- exponent)))
+
;;; Compute an integer N such that 1 <= |2^N * x| < 2.
;;; For the special cases, the following values are used:
;;; x logb
(defun logb (x)
(declare (type double-float x))
(cond ((float-nan-p x)
- x)
- ((float-infinity-p x)
- sb!ext:double-float-positive-infinity)
- ((zerop x)
- ;; The answer is negative infinity, but we are supposed to
- ;; signal divide-by-zero.
- ;; (error 'division-by-zero :operation 'logb :operands (list x))
- (/ -1.0d0 x)
- )
- (t
- (multiple-value-bind (signif expon sign)
- (decode-float x)
- (declare (ignore signif sign))
- ;; DECODE-FLOAT is almost right, except that the exponent
- ;; is off by one.
- (1- expon)))))
+ 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
+ (logb-finite x))))
;;; This function is used to create a complex number of the
;;; appropriate type:
;;; 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))
- (if (subtypep (type-of (realpart z)) 'double-float)
+ (number z))
+ (if (typep (realpart z) 'double-float)
(complex x y)
- ;; Convert anything that's not a DOUBLE-FLOAT to a SINGLE-FLOAT.
- (complex (float x 1.0)
- (float y 1.0))))
+ ;; 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))))
;;; 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)
- ;; Save all FP flags
(let ((x (float (realpart z) 1d0))
- (y (float (imagpart z) 1d0))
- (k 0)
- (rho 0d0))
- (declare (double-float x y)
- (type (double-float 0d0) rho)
- (fixnum k))
+ (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
;; signal would have been raised.
(with-float-traps-masked (:underflow :overflow)
- (setf rho (+ (square x) (square y)))
+ (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))))
- (setf rho sb!ext:double-float-positive-infinity))
- ((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))))
- (setf k (logb (max (abs x) (abs y))))
- (setf rho (+ (square (scalb x (- k)))
- (square (scalb y (- k))))))))
- (values rho k)))
+ (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)
+ (load-time-value
+ #!-long-float
+ (sb!kernel:make-double-float #x1fffff #xfffffffe)
+ #!+long-float
+ (error "(/ least-positive-long-float long-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))))
+ ;; If we're here, neither x nor y are infinity and at
+ ;; least one is non-zero.. Thus logb returns a nice
+ ;; integer.
+ (let ((k (- (logb-finite (max (abs x) (abs y))))))
+ (values (+ (square (scalb x k))
+ (square (scalb y k)))
+ (- k))))
+ (t
+ (values rho 0)))))))
;;; 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 (double-float 0d0) rho)
- (fixnum k))
+ (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
+ ;; space 0 to get maybe-inline functions inlined.
+ (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))))
- (coerce-to-complex-type eta nu z))))
-
+ (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 (number z)
- (fixnum j))
+ (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
;; 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)))
+ (let ((t0 (load-time-value
+ #!-long-float
+ (sb!kernel:make-double-float #x3fe6a09e #x667f3bcd)
+ #!+long-float
+ (error "(/ (sqrt 2l0))")))
+ ;; KLUDGE: if repeatable fasls start failing under some weird
+ ;; xc host, this 1.2d0 might be a good place to examine: while
+ ;; it _should_ be the same in all vaguely-IEEE754 hosts, 1.2
+ ;; is not exactly representable, so something could go wrong.
+ (t1 1.2d0)
+ (t2 3d0)
+ (ln2 (load-time-value
+ #!-long-float
+ (sb!kernel:make-double-float #x3fe62e42 #xfefa39ef)
+ #!+long-float
+ (error "(log 2l0)")))
+ (x (float (realpart z) 1.0d0))
+ (y (float (imagpart z) 1.0d0)))
(multiple-value-bind (rho k)
- (cssqs z)
- (declare (type (double-float 0d0) rho)
- (fixnum k))
+ (cssqs z)
+ (declare (optimize (speed 3)))
(let ((beta (max (abs x) (abs y)))
- (theta (min (abs x) (abs y))))
- (declare (type (double-float 0d0) beta theta))
- (if (and (zerop k)
- (< t0 beta)
- (or (<= beta t1)
- (< rho t2)))
- (setf rho (/ (%log1p (+ (* (- beta 1.0d0)
- (+ beta 1.0d0))
- (* theta theta)))
- 2d0))
- (setf rho (+ (/ (log rho) 2d0)
- (* (+ k j) ln2))))
- (setf theta (atan y x))
- (coerce-to-complex-type rho theta z)))))
+ (theta (min (abs x) (abs y))))
+ (coerce-to-complex-type (if (and (zerop k)
+ (< t0 beta)
+ (or (<= beta t1)
+ (< rho t2)))
+ (/ (%log1p (+ (* (- beta 1.0d0)
+ (+ beta 1.0d0))
+ (* theta theta)))
+ 2d0)
+ (+ (/ (log rho) 2d0)
+ (* (+ k j) ln2)))
+ (atan y x)
+ z)))))
;;; log of Z = log |Z| + i * arg Z
;;;
;;; 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
;;; 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
;;; 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)))
- (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))
- (declare (double-float theta rho half-pi rp beta y eta nu)
- (type (double-float 0d0) x))
+ (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))
+ ;; 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 eta (float-sign y half-pi))
- ;; nu 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))))
- (declare (double-float r d))
- (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)))
- (declare (type (double-float 0d0) t1 t2))
- #+nil
- (setf eta (log (/ (sqrt (sqrt t1)))
- (sqrt t2)))
- (setf eta (* 0.5d0 (log (the (double-float 0.0d0)
- (/ (sqrt t1) t2)))))
- (setf nu (* 0.5d0
- (float-sign y
- (+ half-pi (atan (* 0.5d0 t2))))))))
- (t
- (let ((t1 (+ (abs y) rho)))
- (declare (double-float t1))
- ;; 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))))))))
+ (> (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))))))))
(coerce-to-complex-type (* beta eta)
- (- (* beta nu))
- z)))
+ (- (* beta nu))
+ z))))
;;; 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)))
- (declare (double-float x y))
+ (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))
- (complex (float-sign x)
- (float-sign y 0.0d0)))
- (t
- (let* ((tv (%tan y))
- (beta (+ 1.0d0 (* tv tv)))
- (s (sinh x))
- (rho (sqrt (+ 1.0d0 (* s s)))))
- (declare (double-float tv s)
- (type (double-float 0.0d0) beta rho))
- (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))))))))
+ (load-time-value
+ #!-long-float
+ (sb!kernel:make-double-float #x406633ce #x8fb9f87e)
+ #!+long-float
+ (error "(/ (+ (log 2l0) (log most-positive-long-float)) 4l0)")))
+ (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.
;;;
;;
;; 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))))
+ (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))
;;;
;;; 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))))
+ (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.
;;;
;;; 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))))
+ (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)).
;;;
;;; 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)))
+ (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.
(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)))
+ (result (complex-atanh iz)))
(complex (imagpart result)
- (- (realpart result)))))
+ (- (realpart result)))))
;;; Compute tan z = -i * tanh(i * z)
;;;
;;; 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)))
+ (result (complex-tanh iz)))
(complex (imagpart result)
- (- (realpart result)))))
+ (- (realpart result)))))
projects / sbcl.git / blobdiff