\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.
-;;;
-;;; FIXME: This should be (EVAL-WHEN (COMPILE-EVAL) (SB!XC:DEFMACRO ..)),
-;;; I think.
-(defmacro def-math-rtn (name num-args)
- (let ((function (intern (concatenate 'simple-string
- "%"
- (string-upcase name)))))
+;;; 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))))
`(progn
- (proclaim '(inline ,function))
- (let ((sb!int::*rogue-export* "DEF-MATH-RTN"))
- (export ',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)))))))
-
-(eval-when (:compile-toplevel :load-toplevel :execute)
+ (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))
+ 'double-float)
+ results)))))))
(defun handle-reals (function var)
`((((foreach fixnum single-float bignum ratio))
) ; EVAL-WHEN
\f
;;;; stubs for the Unix math library
+;;;;
+;;;; Many of these are unnecessary on the X86 because they're built
+;;;; into the FPU.
-;;; Please refer to the Unix man pages for details about these routines.
-
-;;; Trigonometric.
+;;; trigonometric
#!-x86 (def-math-rtn "sin" 1)
#!-x86 (def-math-rtn "cos" 1)
#!-x86 (def-math-rtn "tan" 1)
(def-math-rtn "acosh" 1)
(def-math-rtn "atanh" 1)
-;;; Exponential and Logarithmic.
+;;; exponential and logarithmic
#!-x86 (def-math-rtn "exp" 1)
#!-x86 (def-math-rtn "log" 1)
#!-x86 (def-math-rtn "log10" 1)
#!-x86 (def-math-rtn "sqrt" 1)
(def-math-rtn "hypot" 2)
#!-(or hpux x86) (def-math-rtn "log1p" 1)
-
-#!+x86 ;; These are needed for use by byte-compiled files.
-(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
\f
;;;; power functions
;;; 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.
+;;; 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)
-;;; This function precisely calculates base raised to an integral power. It
-;;; separates the cases by the sign of power, for efficiency reasons, as powers
-;;; can be calculated more efficiently if power is a positive integer. Values
-;;; of power are calculated as positive integers, and inverted if negative.
+;;; This function precisely calculates base raised to an integral
+;;; power. It separates the cases by the sign of power, for efficiency
+;;; reasons, as powers can be calculated more efficiently if power is
+;;; 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
(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 off,
-;;; assuming it will return 0 if the result may be complex. If so, we call
-;;; COMPLEX-POW which directly computes the complex result. We also separate
-;;; the complex-real and real-complex cases from the general complex case.
+;;; floating point stuff. If both args are real, we try %POW right
+;;; off, assuming it will return 0 if the result may be complex. If
+;;; so, we call COMPLEX-POW which directly computes the complex
+;;; result. We also separate the complex-real and real-complex cases
+;;; 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))
(labels (;; determine if the double float is an integer.
(let ((pow (sb!kernel::%pow abs-x y)))
(declare (double-float pow))
(case yisint
- (1 ; Odd
+ (1 ; odd
(coerce (* -1d0 pow) rtype))
- (2 ; Even
+ (2 ; even
(coerce pow rtype))
- (t ; Non-integer
+ (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)))))))))))))
+ (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)
(* 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
(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))
(defun phase (number)
#!+sb-doc
- "Returns the angle part of the polar representation of a complex number.
+ "Return the angle part of the polar representation of a complex number.
For complex numbers, this is (atan (imagpart number) (realpart number)).
For non-complex positive numbers, this is 0. For non-complex negative
numbers this is PI."
(defun cis (theta)
#!+sb-doc
- "Return cos(Theta) + i sin(Theta), AKA exp(i Theta)."
- (if (complexp theta)
- (error "Argument to CIS is complex: ~S" theta)
- (complex (cos theta) (sin theta))))
+ "Return cos(Theta) + i sin(Theta), i.e. exp(i Theta)."
+ (declare (type real theta))
+ (complex (cos theta) (sin theta)))
(defun asin (number)
#!+sb-doc
(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.
-
-#+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
(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))
-
(defun cosh (number)
#!+sb-doc
"Return the hyperbolic cosine of NUMBER."
((complex)
(complex-atanh number))))
-;;; HP-UX does not supply a C version of log1p, so
-;;; use the definition.
-
+;;; 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
#!-sb-fluid (declaim (inline %log1p))
#!+hpux
(declare (double-float number)
(optimize (speed 3) (safety 0)))
(the double-float (log (the (double-float 0d0) (+ number 1d0)))))
+\f
+;;;; 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 standard special function system.)
+;;;;
+;;;; This is a set of routines that implement many elementary
+;;;; transcendental functions as specified by ANSI Common Lisp. The
+;;;; implementation is based on Kahan's paper.
+;;;;
+;;;; I believe I have accurately implemented the routines and are
+;;;; correct, but you may want to check for your self.
+;;;;
+;;;; These functions are written for CMU Lisp and take advantage of
+;;;; some of the features available there. It may be possible,
+;;;; however, to port this to other Lisps.
+;;;;
+;;;; Some functions are significantly more accurate than the original
+;;;; definitions in CMU Lisp. In fact, some functions in CMU Lisp
+;;;; give the wrong answer like (acos #c(-2.0 0.0)), where the true
+;;;; answer is pi + i*log(2-sqrt(3)).
+;;;;
+;;;; All of the implemented functions will take any number for an
+;;;; input, but the result will always be a either a complex
+;;;; single-float or a complex double-float.
+;;;;
+;;;; general functions:
+;;;; complex-sqrt
+;;;; complex-log
+;;;; complex-atanh
+;;;; complex-tanh
+;;;; complex-acos
+;;;; complex-acosh
+;;;; complex-asin
+;;;; complex-asinh
+;;;; complex-atan
+;;;; complex-tan
+;;;;
+;;;; utility functions:
+;;;; scalb logb
+;;;;
+;;;; internal functions:
+;;;; square coerce-to-complex-type cssqs complex-log-scaled
+;;;;
+;;;; references:
+;;;; Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much
+;;;; Ado About Nothing's Sign Bit" in Iserles and Powell (eds.) "The
+;;;; State of the Art in Numerical Analysis", pp. 165-211, Clarendon
+;;;; Press, 1987
+;;;;
+;;;; The original CMU CL code requested:
+;;;; Please send any bug reports, comments, or improvements to
+;;;; 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
+;;; constants at cross-compile time, because the cross-compilation
+;;; host might not have support for floating point infinities. Thus,
+;;; 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?
+(declaim (inline square))
+(defun square (x)
+ (declare (double-float x))
+ (* x x))
+
+;;; original CMU CL comment, apparently re. SCALB and LOGB and
+;;; perhaps CSSQS:
+;;; If you have these functions in libm, perhaps they should be used
+;;; instead of these Lisp versions. These versions are probably good
+;;; enough, especially since they are portable.
+
+;;; Compute 2^N * X without computing 2^N first. (Use properties of
+;;; the underlying floating-point format.)
+(declaim (inline scalb))
+(defun scalb (x n)
+ (declare (type double-float x)
+ (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
+;;; NaN NaN
+;;; +/- infinity +infinity
+;;; 0 -infinity
+(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, 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:
+;;; Create complex number with real part X and imaginary part Y
+;;; 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"))
+(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)
+ (complex x y)
+ ;; Convert anything that's not a DOUBLE-FLOAT 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"))
+(defun cssqs (z)
+ (let ((x (float (realpart 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
+ ;; signal would have been raised.
+ (with-float-traps-masked (:underflow :overflow)
+ (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))))
+ (values sb!ext:double-float-positive-infinity 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))))
+ ;; 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.
+(defun complex-sqrt (z)
+ (declare (number z))
+ (multiple-value-bind (rho k)
+ (cssqs z)
+ (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))
+ (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))))
+
+ (cond ((oddp k)
+ (setf k (ash k -1)))
+ (t
+ (setf k (1- (ash k -1)))
+ (setf rho (+ rho rho))))
+
+ (setf rho (scalb (sqrt rho) k))
+
+ (setf eta rho)
+ (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)))))
+
+;;; 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))
+ ;; 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)))
+ (multiple-value-bind (rho k)
+ (cssqs z)
+ (declare (optimize (speed 3)))
+ (let ((beta (max (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)))
+ (/ (%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))
+ (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))
+ (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))
+ ;; 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))))
+ (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))))
+
+;;; Compute tanh z = sinh z / cosh z.
+(defun complex-tanh (z)
+ (declare (number 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)
+ ;; 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.
+;;;
+;;; Z may be any NUMBER, but the result is always a COMPLEX.
+(defun complex-acos (z)
+ ;; Kahan says we should only compute the parts needed. Thus, the
+ ;; REALPART's below should only compute the real part, not the whole
+ ;; complex expression. Doing this can be important because we may get
+ ;; spurious signals that occur in the part that we are not using.
+ ;;
+ ;; However, we take a pragmatic approach and just use the whole
+ ;; expression.
+ ;;
+ ;; NOTE: The formula given by Kahan is somewhat ambiguous in whether
+ ;; it's the conjugate of the square root or the square root of the
+ ;; conjugate. This needs to be checked.
+ ;;
+ ;; I checked. It doesn't matter because (conjugate (sqrt z)) is the
+ ;; same as (sqrt (conjugate z)) for all z. This follows because
+ ;;
+ ;; (conjugate (sqrt z)) = exp(0.5*log |z|)*exp(-0.5*j*arg z).
+ ;;
+ ;; (sqrt (conjugate z)) = exp(0.5*log|z|)*exp(0.5*j*arg conj 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))
+ (let ((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)))))))
+
+;;; 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))
+ (let ((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))))))))
+
+;;; 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))
+ (let ((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)))))))
+
+;;; 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))
+ ;; asinh z = -i * asin (i*z)
+ (let* ((iz (complex (- (imagpart z)) (realpart z)))
+ (result (complex-asin iz)))
+ (complex (imagpart 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))
+ ;; atan z = -i * atanh (i*z)
+ (let* ((iz (complex (- (imagpart z)) (realpart z)))
+ (result (complex-atanh iz)))
+ (complex (imagpart 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))
+ ;; tan z = -i * tanh(i*z)
+ (let* ((iz (complex (- (imagpart z)) (realpart z)))
+ (result (complex-tanh iz)))
+ (complex (imagpart result)
+ (- (realpart result)))))
projects / sbcl.git / blobdiff