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- Log ----------------------------------------------------------------- commit 6127a79e0d6300bc7ac7754190fbd559d00e7401 Author: Raymond Toy toy.raymond@gmail.com Date: Sun Nov 24 10:41:27 2013 -0800
Fix ticket:5
o Implement better pi reduction and also always do pi reduction. o Add test for sin(pi) and cos(2^120), to test pi reduction. o Accuracy for erfc is only 198 bits.
diff --git a/qd-fun.lisp b/qd-fun.lisp index d131a7a..91c0b32 100644 --- a/qd-fun.lisp +++ b/qd-fun.lisp @@ -696,38 +696,46 @@ is the cosine of A" (values mod f) (values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+))))))
-(defun rem-pi/2-int (qd) - ;; Compute qd rem pi/2 = k*pi/2+y. So we compute k + y*2/pi = - ;; qd*2/pi. - ;; - ;; First convert qd to 2^e*I. We already have 2/pi in the form - ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the - ;; integer and fractional parts of this. For the integer part we - ;; only need it modulo 4, because of the periodicity. For the - ;; fractional part, we only need 212 (or so bits of fraction). - ;; - ;; FIXME: But we don't really need to compute all the bits of I*J. - ;; In the product, we really only need the 2 bits to the left of the - ;; binary point, and then 212 bits to the right. This doesn't - ;; require doing the full multiplication. - (multiple-value-bind (i e s) +(defun split-prod-2/pi (qd) + ;; Compute k and f such that k + f = qd*2/pi, where k is an intege + ;; and f < 1. Since we do not need more than the the least + ;; two bits from k, only the least two bits of k are returned. + (multiple-value-bind (i e) (integer-decode-qd qd) + ;; First convert qd to 2^e*I. We already have 2/pi in the form + ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the + ;; integer and fractional parts of this. For the integer part we + ;; only need it modulo 4, because of the periodicity. For the + ;; fractional bits, we return the actual fraction as a ratio. + ;; + ;; FIXME: We compute the entire product I*J, but we only need the + ;; 2 bits to the left of the binary point. But we need many of + ;; the bits to the right of the binary point becaue we have no + ;; lower bound on the difference between a quad-double and pi. (let* ((exp (- e (integer-length +2/pi-bits+))) - (prod (* (* s i) +2/pi-bits+)) - (mod (ldb (byte 2 (- exp)) prod)) - ;; A quad-double has about 212 bits, but we add another 53 - ;; (5 doubles) for some extra accuracty. - (qd-bits 265) - (frac (ldb (byte qd-bits (- (- exp) qd-bits)) prod)) - (f (mul-qd (scale-float-qd (rational-to-qd frac) (- qd-bits)) - +qd-pi/2+))) - ;; We want the remainder part to be <= pi/4 because the trig - ;; functions want that. So if the fraction is too big, adjust - ;; it, and mod value. - (if (qd-<= (abs-qd f) +qd-pi/4+) - (values mod f) - (values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+)))))) + (binary-point (- exp)) + (prod (* i +2/pi-bits+)) + (int (ldb (byte 2 binary-point) prod)) + (frac (/ (ldb (byte binary-point 0) prod) + (ash 1 binary-point)))) + (values int frac))))
+(defun rem-pi/2-int (qd) + "Compute qd rem pi/2 = k + f, where k is an integer and |f| < + pi/4. Two values are returned: k mod 4 and f." + (if (minusp-qd qd) + (multiple-value-bind (k f) + (rem-pi/2-int (neg-qd qd)) + (values (mod (- k) 4) (neg-qd f))) + (multiple-value-bind (k f) + (split-prod-2/pi qd) + ;; The trig functions want the arg to be less than pi/4. That + ;; is f < 1/2. Thus adjust k and f accordingly. + (when (> f 1/2) + (setf k (mod (1+ k) 4)) + (setf f (- f 1))) + (values k (mul-qd +qd-pi/2+ (rational-to-qd f)))))) + (defun rem-pi/2-int-b (qd) (declare (type %quad-double qd)) (multiple-value-bind (i e s) @@ -785,6 +793,13 @@ is the cosine of A" (sub-qd a (mul-qd-d +qd-pi/2+ (float quot 1d0)))))) (t (rem-pi/2-int a)))) + +(defun rem-pi/2 (a) + ;; If the number is small enough, we don't need to use the full + ;; precision algorithm to compute the remainder. The value of 1024 + ;; here is rather arbitrary. We should do an analysis to figure + ;; where the breakpoint should be. + (rem-pi/2-int a))
(defun sin-qd (a) diff --git a/rt-tests.lisp b/rt-tests.lisp index caf568f..69f6ee7 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -187,6 +187,18 @@ (check-accuracy 212 val true)) nil)
+;;; sin +(rt:deftest oct.sin.pi + (not (zerop (sin +pi+))) + t) + +;;; cos +(rt:deftest oct.cos.big + (let* ((val (cos (scale-float #q1 120))) + (err (abs (- val -0.9258790228548379d0)))) + (<= err 5.2d-17)) + t) + ;;; Tests of atan where we know the analytical result (rt:deftest oct.atan.1 (let* ((arg (/ (sqrt #q3))) @@ -1731,6 +1743,6 @@ nil)
(rt:deftest erfc - (check-accuracy 210 (erfc #q-4) + (check-accuracy 198 (erfc #q-4) #q1.9999999845827420997199811478403265131159514278547464108088316570950057869589732) nil)
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Summary of changes: qd-fun.lisp | 73 ++++++++++++++++++++++++++++++++++----------------------- rt-tests.lisp | 14 ++++++++++- 2 files changed, 57 insertions(+), 30 deletions(-)
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