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- 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.
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Summary of changes: src/code/irrat.lisp | 434 --------------------------------------------------- 1 file changed, 434 deletions(-)
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