This is an automated email from the git hooks/post-receive script. It was generated because a ref change was pushed to the repository containing the project "CMU Common Lisp". The branch, master has been updated via 7107249f265148e246a9eec84b15cfbe96121594 (commit) from 13f320d9568705f58010d456b957917b2d2c62c6 (commit) Those revisions listed above that are new to this repository have not appeared on any other notification email; so we list those revisions in full, below. - Log ----------------------------------------------------------------- commit 7107249f265148e246a9eec84b15cfbe96121594 Author: Raymond Toy <toy.raymond@gmail.com> Date: Thu Jul 31 15:50:44 2014 -0700 Finally remove the Lisp implementation of the trig functions that are now in C. diff --git a/src/code/irrat.lisp b/src/code/irrat.lisp index c48d09d..7c8b031 100644 --- a/src/code/irrat.lisp +++ b/src/code/irrat.lisp @@ -208,440 +208,6 @@ (declare (ignore ign)) (values s c))) -#|| -;; Implement sin/cos/tan in Lisp. These are based on the routines -;; from fdlibm. - -;; Block compile so the trig routines don't cons their args when -;; calling the kernel trig routines. -(declaim (ext:start-block kernel-sin kernel-cos kernel-tan - %sin %cos %tan - %sincos)) - -;; kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 -;; Input x is assumed to be bounded by ~pi/4 in magnitude. -;; Input y is the tail of x. -;; Input iy indicates whether y is 0. (if iy=0, y assume to be 0). -;; -;; Algorithm -;; 1. Since sin(-x) = -sin(x), we need only to consider positive x. -;; 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. -;; 3. sin(x) is approximated by a polynomial of degree 13 on -;; [0,pi/4] -;; 3 13 -;; sin(x) ~ x + S1*x + ... + S6*x -;; where -;; -;; |sin(x) 2 4 6 8 10 12 | -58 -;; |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 -;; | x | -;; -;; 4. sin(x+y) = sin(x) + sin'(x')*y -;; ~ sin(x) + (1-x*x/2)*y -;; For better accuracy, let -;; 3 2 2 2 2 -;; r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) -;; then 3 2 -;; sin(x) = x + (S1*x + (x *(r-y/2)+y)) - -(declaim (ftype (function (double-float double-float fixnum) - double-float) - kernel-sin)) - -(defun kernel-sin (x y iy) - (declare (type (double-float -1d0 1d0) x y) - (fixnum iy) - (optimize (speed 3) (safety 0))) - (let ((ix (ldb (byte 31 0) (kernel:double-float-high-bits x)))) - (when (< ix #x3e400000) - ;; |x| < 2^-27 - ;; Signal inexact if x /= 0 - (if (zerop (truncate x)) - (return-from kernel-sin x) - (return-from kernel-sin x))) - (let* ((s1 -1.66666666666666324348d-01) ; #xBFC55555 #x55555549 - (s2 8.33333333332248946124d-03) ; #x3F811111 #x1110F8A6 - (s3 -1.98412698298579493134d-04) ; #xBF2A01A0 #x19C161D5 - (s4 2.75573137070700676789d-06) ; #x3EC71DE3 #x57B1FE7D - (s5 -2.50507602534068634195d-08) ; #xBE5AE5E6 #x8A2B9CEB - (s6 1.58969099521155010221d-10) ; #x3DE5D93A #x5ACFD57C - (z (* x x)) - (v (* z x)) - (r (+ s2 - (* z - (+ s3 - (* z - (+ s4 - (* z - (+ s5 - (* z s6)))))))))) - (if (zerop iy) - (+ x (* v (+ s1 (* z r)))) - (- x (- (- (* z (- (* .5 y) - (* v r))) - y) - (* v s1))))))) - -;; kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 -;; Input x is assumed to be bounded by ~pi/4 in magnitude. -;; Input y is the tail of x. -;; -;; Algorithm -;; 1. Since cos(-x) = cos(x), we need only to consider positive x. -;; 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. -;; 3. cos(x) is approximated by a polynomial of degree 14 on -;; [0,pi/4] -;; 4 14 -;; cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x -;; where the remez error is -;; -;; | 2 4 6 8 10 12 14 | -58 -;; |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 -;; | | -;; -;; 4 6 8 10 12 14 -;; 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then -;; cos(x) = 1 - x*x/2 + r -;; since cos(x+y) ~ cos(x) - sin(x)*y -;; ~ cos(x) - x*y, -;; a correction term is necessary in cos(x) and hence -;; cos(x+y) = 1 - (x*x/2 - (r - x*y)) -;; For better accuracy when x > 0.3, let qx = |x|/4 with -;; the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. -;; Then -;; cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). -;; Note that 1-qx and (x*x/2-qx) is EXACT here, and the -;; magnitude of the latter is at least a quarter of x*x/2, -;; thus, reducing the rounding error in the subtraction. -(declaim (ftype (function (double-float double-float) - double-float) - kernel-cos)) - -(defun kernel-cos (x y) - (declare (type (double-float -1d0 1d0) x y) - (optimize (speed 3) (safety 0))) - ;; cos(-x) = cos(x), so we just compute cos(|x|). - (let ((ix (ldb (byte 31 0) (kernel:double-float-high-bits x)))) - ;; cos(x) = 1 when |x| < 2^-27 - (when (< ix #x3e400000) - ;; Signal inexact if x /= 0 - (if (zerop (truncate x)) - (return-from kernel-cos 1d0) - (return-from kernel-cos 1d0))) - (let* ((c1 4.16666666666666019037d-02) - (c2 -1.38888888888741095749d-03) - (c3 2.48015872894767294178d-05) - (c4 -2.75573143513906633035d-07) - (c5 2.08757232129817482790d-09) - (c6 -1.13596475577881948265d-11) - (z (* x x)) - (r (* z - (+ c1 - (* z - (+ c2 - (* z - (+ c3 - (* z - (+ c4 - (* z - (+ c5 - (* z c6))))))))))))) - (cond ((< ix #x3fd33333) - ;; \x| < 0.3 - (- 1 (- (* .5 z) - (- (* z r) - (* x y))))) - (t - ;; qx = 0.28125 if |x| > 0.78125, else x/4 dropping the - ;; least significant 32 bits. - (let* ((qx (if (> ix #x3fe90000) - 0.28125d0 - ;; x/4, exactly, and also dropping the - ;; least significant 32 bits of the - ;; fraction. - (make-double-float (- ix #x00200000) - 0))) - (hz (- (* 0.5 z) qx)) - (a (- 1 qx))) - (- a (- hz (- (* z r) - (* x y)))))))))) - -(declaim (type (simple-array double-float (*)) tan-coef)) -(defconstant tan-coef - (make-array 13 :element-type 'double-float - :initial-contents - '(3.33333333333334091986d-01 - 1.33333333333201242699d-01 - 5.39682539762260521377d-02 - 2.18694882948595424599d-02 - 8.86323982359930005737d-03 - 3.59207910759131235356d-03 - 1.45620945432529025516d-03 - 5.88041240820264096874d-04 - 2.46463134818469906812d-04 - 7.81794442939557092300d-05 - 7.14072491382608190305d-05 - -1.85586374855275456654d-05 - 2.59073051863633712884d-05))) - -;; kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 -;; Input x is assumed to be bounded by ~pi/4 in magnitude. -;; Input y is the tail of x. -;; Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. -;; -;; Algorithm -;; 1. Since tan(-x) = -tan(x), we need only to consider positive x. -;; 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. -;; 3. tan(x) is approximated by a odd polynomial of degree 27 on -;; [0,0.67434] -;; 3 27 -;; tan(x) ~ x + T1*x + ... + T13*x -;; where -;; -;; |tan(x) 2 4 26 | -59.2 -;; |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 -;; | x | -;; -;; Note: tan(x+y) = tan(x) + tan'(x)*y -;; ~ tan(x) + (1+x*x)*y -;; Therefore, for better accuracy in computing tan(x+y), let -;; 3 2 2 2 2 -;; r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) -;; then -;; 3 2 -;; tan(x+y) = x + (T1*x + (x *(r+y)+y)) -;; -;; 4. For x in [0.67434,pi/4], let y = pi/4 - x, then -;; tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) -;; = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) -(declaim (ftype (function (double-float double-float fixnum) - double-float) - kernel-tan)) - -(defun kernel-tan (x y iy) - (declare (type (double-float -1d0 1d0) x y) - (type (member -1 1) iy) - (optimize (speed 3) (safety 0))) - (let* ((hx (kernel:double-float-high-bits x)) - (ix (logand hx #x7fffffff)) - (w 0d0) - (z 0d0) - (v 0d0) - (s 0d0) - (r 0d0)) - (declare (double-float w z v s r)) - (when (< ix #x3e300000) - ;; |x| < 2^-28 - (when (zerop (truncate x)) - (cond ((zerop (logior (logior ix (kernel:double-float-low-bits x)) - (+ iy 1))) - ;; x = 0 (because hi and low bits are 0) and iy = -1 - ;; (cot) - (return-from kernel-tan (/ (abs x)))) - ((= iy 1) - (return-from kernel-tan x)) - (t - ;; x /= 0 and iy = -1 (cot) - ;; Compute -1/(x+y) carefully - (let ((a 0d0) - (tt 0d0)) - (setf w (+ x y)) - (setf z (make-double-float (double-float-high-bits w) 0)) - (setf v (- y (- z x))) - (setf a (/ -1 w)) - (setf tt (make-double-float (double-float-high-bits a) 0)) - (setf s (+ 1 (* tt z))) - (return-from kernel-tan (+ tt - (* a (+ s (* tt v)))))))))) - (when (>= ix #x3FE59428) - ;; |x| > .6744 - (when (minusp hx) - (setf x (- x)) - (setf y (- y))) - ;; The two constants below are such that pi/4 + pi/4_lo is pi/4 - ;; to twice the accuracy of a double float. - ;; - ;; z = pi/4-x - (setf z (- (make-double-float #x3FE921FB #x54442D18) x)) - ;; w = pi/4_lo - y. - (setf w (- (make-double-float #x3C81A626 #x33145C07) y)) - (setf x (+ z w)) - (setf y 0d0)) - (setf z (* x x)) - (setf w (* z z)) - ;; Break x^5*(T[1]+x^2*T[2]+...) into - ;; x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + - ;; x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) - (setf r (+ (aref tan-coef 1) - (* w - (+ (aref tan-coef 3) - (* w - (+ (aref tan-coef 5) - (* w - (+ (aref tan-coef 7) - (* w - (+ (aref tan-coef 9) - (* w (aref tan-coef 11)))))))))))) - (setf v (* z - (+ (aref tan-coef 2) - (* w - (+ (aref tan-coef 4) - (* w - (+ (aref tan-coef 6) - (* w - (+ (aref tan-coef 8) - (* w - (+ (aref tan-coef 10) - (* w (aref tan-coef 12))))))))))))) - (setf s (* z x)) - (setf r (+ y (* z (+ (* s (+ r v)) - y)))) - (incf r (* s (aref tan-coef 0))) - (setf w (+ x r)) - (when (>= ix #x3FE59428) - (let ((v (float iy 1d0))) - (return-from kernel-tan - (* (- 1 (logand 2 (ash hx -30))) - (- v - (* 2 - (- x (- (/ (* w w) - (+ w v)) - r)))))))) - (when (= iy 1) - (return-from kernel-tan w)) - ;; Compute 1/w=1/(x+r) carefully - (let ((a 0d0) - (tt 0d0)) - (setf z (kernel:make-double-float (kernel:double-float-high-bits w) 0)) - (setf v (- r (- z x))) ; z + v = r + x - (setf a (/ -1 w)) - (setf tt (kernel:make-double-float (kernel:double-float-high-bits a) 0)) - (setf s (+ 1 (* tt z))) - (+ tt - (* a - (+ s (* tt v))))))) - -;; Return sine function of x. -;; -;; kernel function: -;; __kernel_sin ... sine function on [-pi/4,pi/4] -;; __kernel_cos ... cose function on [-pi/4,pi/4] -;; __ieee754_rem_pio2 ... argument reduction routine -;; -;; Method. -;; Let S,C and T denote the sin, cos and tan respectively on -;; [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 -;; in [-pi/4 , +pi/4], and let n = k mod 4. -;; We have -;; -;; n sin(x) cos(x) tan(x) -;; ---------------------------------------------------------- -;; 0 S C T -;; 1 C -S -1/T -;; 2 -S -C T -;; 3 -C S -1/T -;; ---------------------------------------------------------- -;; -;; Special cases: -;; Let trig be any of sin, cos, or tan. -;; trig(+-INF) is NaN, with signals; -;; trig(NaN) is that NaN; -;; -;; Accuracy: -;; TRIG(x) returns trig(x) nearly rounded -(defun %sin (x) - (declare (double-float x) - (optimize (speed 3))) - (let ((ix (ldb (byte 31 0) (kernel:double-float-high-bits x)))) - (cond - ((<= ix #x3fe921fb) - ;; |x| < pi/4, approx - (kernel-sin x 0d0 0)) - ((>= ix #x7ff00000) - ;; sin(Inf or NaN) is NaN - (- x x)) - (t - ;; Argument reduction needed - (multiple-value-bind (n y0 y1) - (%ieee754-rem-pi/2 x) - (case (logand n 3) - (0 - (kernel-sin y0 y1 1)) - (1 - (kernel-cos y0 y1)) - (2 - (- (kernel-sin y0 y1 1))) - (3 - (- (kernel-cos y0 y1))))))))) - -(defun %cos (x) - (declare (double-float x) - (optimize (speed 3))) - (let ((ix (ldb (byte 31 0) (kernel:double-float-high-bits x)))) - (cond - ((< ix #x3fe921fb) - ;;|x| < pi/4, approx - (kernel-cos x 0d0)) - ((>= ix #x7ff00000) - ;; cos(Inf or NaN) is NaN - (- x x)) - (t - ;; Argument reduction needed - (multiple-value-bind (n y0 y1) - (%ieee754-rem-pi/2 x) - (ecase (logand n 3) - (0 - (kernel-cos y0 y1)) - (1 - (- (kernel-sin y0 y1 1))) - (2 - (- (kernel-cos y0 y1))) - (3 - (kernel-sin y0 y1 1)))))))) - -(defun %tan (x) - (declare (double-float x) - (optimize (speed 3))) - (let ((ix (logand #x7fffffff (kernel:double-float-high-bits x)))) - (cond ((<= ix #x3fe921fb) - ;; |x| < pi/4 - (kernel-tan x 0d0 1)) - ((>= ix #x7ff00000) - ;; tan(Inf or Nan) is NaN - (- x x)) - (t - (multiple-value-bind (n y0 y1) - (%ieee754-rem-pi/2 x) - (let ((flag (- 1 (ash (logand n 1) 1)))) - ;; flag = 1 if n even, -1 if n odd - (kernel-tan y0 y1 flag))))))) -;; Compute sin and cos of x, simultaneously. -(defun %sincos (x) - (declare (double-float x) - (optimize (speed 3))) - (cond ((<= (abs x) (/ pi 4)) - (values (kernel-sin x 0d0 0) - (kernel-cos x 0d0))) - (t - ;; Argument reduction needed - (multiple-value-bind (n y0 y1) - (%ieee754-rem-pi/2 x) - (case (logand n 3) - (0 - (values (kernel-sin y0 y1 1) - (kernel-cos y0 y1))) - (1 - (values (kernel-cos y0 y1) - (- (kernel-sin y0 y1 1)))) - (2 - (values (- (kernel-sin y0 y1 1)) - (- (kernel-cos y0 y1)))) - (3 - (values (- (kernel-cos y0 y1)) - (kernel-sin y0 y1 1)))))))) -;;(declaim (ext:end-block)) -||# -  ;;;; Power functions. ----------------------------------------------------------------------- Summary of changes: src/code/irrat.lisp | 434 --------------------------------------------------- 1 file changed, 434 deletions(-) hooks/post-receive -- CMU Common Lisp