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- Log ----------------------------------------------------------------- commit b71b7c854469191e462162c9d11d0c35222a284a Author: Raymond Toy toy.raymond@gmail.com Date: Fri Dec 20 18:12:03 2013 -0800

Increase accuracy of tan near multiples of pi/4.

code/irrat-dd.lisp o Make REDUCE-ARG return an extra result. Since the reduction returns 3 double's, return a double-double result and the third double result for extra accuracy o Update dd-%tan and dd-%%tan to take the extra arg. o Add new constant dd-pi/4-lo. o Increase accuracy of tan by using the relationship tan(pi/4-y)=(1-tan(y))/(1+tan(y)).

tests/trig.lisp: o Update the allowed error threshold for two tests to reflect the increased accuracy.

diff --git a/src/code/irrat-dd.lisp b/src/code/irrat-dd.lisp index 381d678..2eabb58 100644 --- a/src/code/irrat-dd.lisp +++ b/src/code/irrat-dd.lisp @@ -56,6 +56,12 @@ 0.7853981633974483096156608458198757210492923w0 _N"Pi/4")

+;; dd-pi/4-lo is such that dd-pi/4-lo + dd-pi/4 is equal to pi/4 to +;; twice the precision of a double-double-float. +(defconstant dd-pi/4-lo + -7.486924524295848886603985669688687133352026408988280860093283232w-34 + ) + ;; log2-c1 and log-c2 are log(2) arranged in such a way that log2-c1 + ;; log2-c2 is log(2) to an accuracy greater than double-double-float. (defconstant log2-c1 @@ -1072,9 +1078,24 @@ pi/4 11001001000011111101101010100010001000010110100011000 010001101001100010 (/ y) y))))

-(defun dd-%%tan (x) - (declare (type double-double-float x)) - (dd-tancot x nil)) +(defun dd-%%tan (x extra) + (declare (type double-double-float x) + (double-float extra)) + (cond ((>= (abs x) 0.6744) + ;; For 0.6744 <= |x| <= pi/4, we want to use the relationship + ;; + ;; tan(x) = tan(pi/4-y) = (1 - tan(y))/(1 + tan(y)) + ;; = 1 - 2*(tan(y) - tan(y)^2)/(1+tan(y)) + (if (minusp x) + (- (dd-%%tan (- x) (- extra))) + (let* ((z (- dd-pi/4 x)) + (w (- dd-pi/4-lo extra)) + (tan (dd-tancot (+ z w) nil))) + (- 1 + (/ (* 2 (- tan (* tan tan))) + (+ 1 tan)))))) + (t + (dd-tancot x nil))))

(declaim (inline %kernel-rem-pi/2)) (alien:def-alien-routine ("__kernel_rem_pio2" %kernel-rem-pi/2) c-call:int @@ -1139,10 +1160,9 @@ pi/4 11001001000011111101101010100010001000010110100011000 010001101001100010 (length parts) 3 (vector-sap two-over-pi)))) - (sum (+ (coerce (aref y 2) 'double-double-float) - (coerce (aref y 1) 'double-double-float) + (sum (+ (coerce (aref y 1) 'double-double-float) (coerce (aref y 0) 'double-double-float)))) - (values n sum)))) + (values n sum (aref y 2)))))

(declaim (ftype (function (double-double-float) double-double-float) @@ -1181,15 +1201,15 @@ pi/4 11001001000011111101101010100010001000010110100011000 010001101001100010 dd-%tan)) (defun dd-%tan (x) (declare (double-double-float x)) - (cond ((< (abs x) (/ pi 4)) - (dd-%%tan x)) + (cond ((<= (abs x) (/ pi 4)) + (dd-%%tan x 0d0)) (t ;; Argument reduction needed - (multiple-value-bind (n reduced) + (multiple-value-bind (n reduced extra) (reduce-arg x) (if (evenp n) - (dd-%%tan reduced) - (- (/ (dd-%%tan reduced)))))))) + (dd-%%tan reduced extra) + (- (/ (dd-%%tan reduced extra))))))))

(defun dd-%sincos (x) (declare (double-double-float x)) diff --git a/src/tests/trig.lisp b/src/tests/trig.lisp index b46c904..58d5440 100644 --- a/src/tests/trig.lisp +++ b/src/tests/trig.lisp @@ -376,12 +376,12 @@ (assert-eq t (rel-or-abs-error (tan (* 7/4 kernel:dd-pi)) -1.000000000000000000000000000000001844257310064121018312678894979w0 - 6.467w-33)) + 3.422w-49)) ;; Test for argument reduction with n odd (assert-eq t (rel-or-abs-error (tan (* 9/4 kernel:dd-pi)) 1.000000000000000000000000000000025802415787810837455445433037983w0 - 5.773w-33)) + 0w0)) ;; Test for argument reduction, big value (assert-eq t (rel-or-abs-error (tan (scale-float 1w0 120))

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Summary of changes: src/code/irrat-dd.lisp | 42 +++++++++++++++++++++++++++++++----------- src/tests/trig.lisp | 4 ++-- 2 files changed, 33 insertions(+), 13 deletions(-)

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