+
+;; The x86 port used not to reduce the arguments of transcendentals
+;; correctly.
+;; This test is valid only for x86: The x86 port uses the builtin x87
+;; FPU instructions to implement the trigonometric functions; other
+;; ports rely on the system's math library. These two differ in the
+;; precision of pi used for the range reduction and so yield results
+;; that can differ by arbitrarily large amounts for large inputs.
+;; The test expects the x87 results.
+(with-test (:name (:range-reduction :x87)
+ :skipped-on '(not :x86))
+ (flet ((almost= (x y)
+ (< (abs (- x y)) 1d-5)))
+ (macrolet ((foo (op value)
+ `(let ((actual (,op ,value))
+ (expected (,op (mod ,value (* 2 pi)))))
+ (unless (almost= actual expected)
+ (error "Inaccurate result for ~a: expected ~a, got ~a"
+ (list ',op ,value) expected actual)))))
+ (let ((big (* pi (expt 2d0 70)))
+ (mid (coerce most-positive-fixnum 'double-float))
+ (odd (* pi most-positive-fixnum)))
+ (foo sin big)
+ (foo sin mid)
+ (foo sin odd)
+ (foo sin (/ odd 2d0))
+
+ (foo cos big)
+ (foo cos mid)
+ (foo cos odd)
+ (foo cos (/ odd 2d0))
+
+ (foo tan big)
+ (foo tan mid)
+ (foo tan odd)))))
+
+;; To test the range reduction of trigonometric functions we need a much
+;; more accurate approximation of pi than CL:PI is. Calculating this is
+;; more fun than copy-pasting a constant and Gauss-Legendre converges
+;; extremely fast.
+(defun pi-gauss-legendre (n-bits)
+ "Return a rational approximation to pi using the Gauss-Legendre
+algorithm. The calculations are done with integers, representing
+multiples of (expt 2 (- N-BITS)), and the result is an integral multiple
+of this number. The result is accurate to a few less than N-BITS many
+fractional bits."
+ (let ((a (ash 1 n-bits)) ; scaled 1
+ (b (isqrt (expt 2 (1- (* n-bits 2))))) ; scaled (sqrt 1/2)
+ (c (ash 1 (- n-bits 2))) ; scaled 1/4
+ (d 0))
+ (loop
+ (when (<= (- a b) 1)
+ (return))
+ (let ((a1 (ash (+ a b) -1)))
+ (psetf a a1
+ b (isqrt (* a b))
+ c (- c (ash (expt (- a a1) 2) (- d n-bits)))
+ d (1+ d))))
+ (/ (round (expt (+ a b) 2) (* 4 c))
+ (ash 1 n-bits))))
+
+;; Test that the range reduction of trigonometric functions is done
+;; with a sufficiently accurate value of pi that the reduced argument
+;; is correct to nearly double-float precision even for arguments of
+;; very large absolute value.
+;; This test is skipped on x86; as to why see the comment at the test
+;; (:range-reduction :x87) above.
+(with-test (:name (:range-reduction :precise-pi)
+ :skipped-on :x86)
+ (let ((rational-2pi (* 2 (pi-gauss-legendre 2200))))
+ (labels ((mod-2pi (x)
+ "Return X modulo 2 pi, where pi is precise enough that the
+ result is exact to double-float precision for all possible
+ double-float arguments."
+ (declare (type double-float x))
+ (coerce (mod (rational x) rational-2pi)
+ 'double-float))
+ (test (op x)
+ (let ((actual (funcall op x))
+ (expected (funcall op (mod-2pi x))))
+ ;; Some of the test values are chosen to reduce modulo
+ ;; 2 pi to small numbers (between 1d-10 and 1d-7),
+ ;; making their sine and tangent this small, too.
+ ;; For other test values the absolute value of the
+ ;; tangent may be much larger than 1. Therefore we
+ ;; measure relative instead of absolute error.
+ (unless (or (= actual expected 0)
+ (and (= (signum actual) (signum expected))
+ (< (abs (/ (- actual expected)
+ (+ actual expected)))
+ (/ 1d-12 2))))
+ (error "Inaccurate result for ~a: expected ~a, got ~a"
+ (list op x) expected actual)))))
+ (dolist (op '(sin cos tan))
+ (dolist (val `(,(coerce most-positive-fixnum 'double-float)
+ ,@(loop for v = most-positive-double-float
+ then (expt v 4/5)
+ while (> v (expt 2 50))
+ collect v)
+ 6.543554061677196d28
+ 1.5334254929660437d43
+ 1.837213298702053d93
+ 4.913896894631919d229))
+ (test op val))))))
projects / sbcl.git / blobdiff