;;; HP-UX does not supply a C version of log1p, so
;;; use the definition.
-
#!+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
+;;;; 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 toy@rtp.ericsson.se.
+
+;;; 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))
+(declaim (ftype (function (double-float) (double-float 0d0)) square))
+(defun square (x)
+ (declare (double-float x)
+ (values (double-float 0d0)))
+ (* 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))
+
+;;; 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.
+ ;; (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)))))
+
+;;; 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 1.0)
+ (float y 1.0))))
+
+;;; 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)
+ ;; 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))
+ ;; 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)))
+ (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)))
+
+;;; 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 (double-float 0d0) rho)
+ (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))
+
+ (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 (type (double-float 0d0) rho)
+ (fixnum k))
+ (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)))))
+
+;;; 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))
+ (declare (double-float theta rho half-pi rp beta y eta nu)
+ (type (double-float 0d0) x))
+ (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))))))))
+ (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)))
+ (declare (double-float x y))
+ (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))))))))
+
+;;; 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)))))
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