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;;;; Copyright (c) 2007,2011 Raymond Toy
;;;;
;;;; Permission is hereby granted, free of charge, to any person
;;;; obtaining a copy of this software and associated documentation
;;;; files (the "Software"), to deal in the Software without
;;;; restriction, including without limitation the rights to use,
;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
;;;; copies of the Software, and to permit persons to whom the
;;;; Software is furnished to do so, subject to the following
;;;; conditions:
;;;;
;;;; The above copyright notice and this permission notice shall be
;;;; included in all copies or substantial portions of the Software.
;;;;
;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
;;;; OTHER DEALINGS IN THE SOFTWARE.
(eval-when (:compile-toplevel :load-toplevel :execute)
(setf *readtable* *oct-readtable*))
;; For the tests, we need to turn off underflow for clisp.
#+clisp
(ext:without-package-lock ()
(setq sys::*inhibit-floating-point-underflow* t))
;; Compute how many bits are the same for two numbers EST and TRUE.
;; Return T if they are identical.
(defun bit-accuracy (est true)
(let* ((diff (abs (- est true)))
(err (float (if (zerop true)
diff
(/ diff (abs true)))
1d0)))
(if (zerop diff)
t
(- (log err 2)))))
;; Check actual value EST is with LIMIT bits of the true value TRUE.
;; If so, return NIL. Otherwise, return a list of the actual bits of
;; accuracy, the desired accuracy, and the values. This is mostly to
;; make it easy to see what the actual accuracy was and the arguments
;; for the test, which is important for the tests that use random
;; values.
(defun check-accuracy (limit est true)
(let ((bits (bit-accuracy est true)))
(if (not (eq bits t))
(if (and (not (float-nan-p (realpart est)))
(not (float-nan-p bits))
(< bits limit))
(list bits limit est true)))))
(defvar *null* (make-broadcast-stream))
;;; Some simple tests from the Yozo Hida's qd package.
(rt:deftest float.1
(float 3/2)
1.5)
(rt:deftest float.2
(float 3/2 1d0)
1.5d0)
(rt:deftest float.3
(float 1.5d0)
1.5d0)
(rt:deftest float.4
(= (float #q1.5) #q1.5)
t)
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(rt:deftest ceiling-d.1
(multiple-value-list (ceiling -50d0))
(-50 0d0))
(rt:deftest ceiling-d.2
(let ((z -50.1d0))
(multiple-value-bind (res rem)
(ceiling -50.1d0)
(list res (= z (+ res rem)))))
(-50 t))
(rt:deftest ceiling-q.1
(multiple-value-bind (res rem)
(ceiling #q-50q0)
(list res (zerop rem)))
(-50 t))
(rt:deftest ceiling-q.2
(let ((z #q-50.1q0))
(multiple-value-bind (res rem)
(ceiling z)
(list res (= z (+ res rem)))))
(-50 t))
(rt:deftest truncate-d.1
(multiple-value-list (truncate -50d0))
(-50 0d0))
(rt:deftest truncate-q.1
(multiple-value-bind (res rem)
(truncate #q-50q0)
(list res (zerop rem)))
(-50 t))
(rt:deftest fceiling-d.1
(multiple-value-list (fceiling -50d0))
(-50d0 0d0))
(rt:deftest fceiling-d.2
(let ((z -50.1d0))
(multiple-value-bind (res rem)
(fceiling -50.1d0)
(list res (= z (+ res rem)))))
(-50d0 t))
(rt:deftest fceiling-q.1
(multiple-value-bind (res rem)
(fceiling #q-50q0)
(list (= res -50) (zerop rem)))
(t t))
(rt:deftest fceiling-q.2
(let ((z #q-50.1q0))
(multiple-value-bind (res rem)
(fceiling z)
(list (= res -50) (= z (+ res rem)))))
(t t))
(rt:deftest ftruncate-d.1
(multiple-value-list (ftruncate -50d0))
(-50d0 0d0))
(rt:deftest ftruncate-q.1
(multiple-value-bind (res rem)
(ftruncate #q-50q0)
(list (= res -50) (zerop rem)))
(t t))
;; Pi via Machin's formula
(rt:deftest oct.pi.machin
(let* ((*standard-output* *null*)
(val (make-instance 'qd-real :value (octi::test2 nil)))
(check-accuracy 213 val true))
nil)
;; Pi via Salamin-Brent algorithm
(rt:deftest oct.pi.salamin-brent
(let* ((*standard-output* *null*)
(val (make-instance 'qd-real :value (octi::test3 nil)))
(check-accuracy 202 val true))
nil)
;; Pi via Borweign's Quartic formula
(rt:deftest oct.pi.borweign
(let* ((*standard-output* *null*)
(val (make-instance 'qd-real :value (octi::test4 nil)))
(check-accuracy 211 val true))
nil)
;; e via Taylor series
(rt:deftest oct.e.taylor
(let* ((*standard-output* *null*)
(val (make-instance 'qd-real :value (octi::test5 nil)))
(true (make-instance 'qd-real :value octi::+qd-e+)))
(check-accuracy 212 val true))
nil)
;; log(2) via Taylor series
(rt:deftest oct.log2.taylor
(let* ((*standard-output* *null*)
(val (make-instance 'qd-real :value (octi::test6 nil)))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
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(check-accuracy 212 val true))
nil)
;;; Tests of atan where we know the analytical result
(rt:deftest oct.atan.1
(let* ((arg (/ (sqrt #q3)))
(y (/ (atan arg) +pi+))
(true (/ #q6)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan.2
(let* ((arg (sqrt #q3))
(y (/ (atan arg) +pi+))
(true (/ #q3)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan.3
(let* ((arg #q1)
(y (/ (atan arg) +pi+))
(true (/ #q4)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan.4
(let* ((arg #q1q100)
(y (/ (atan arg) +pi+))
(true #q.5))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan.5
(let* ((arg #q-1q100)
(y (/ (atan arg) +pi+))
(true #q-.5))
(check-accuracy 212 y true))
nil)
(defun atan-qd/duplication (arg)
(make-instance 'qd-real
:value (octi::atan-qd/duplication (qd-value arg))))
;;; Tests of atan where we know the analytical result. Same tests,
;;; but using the atan duplication formula.
(rt:deftest oct.atan/dup.1
(let* ((arg (/ (sqrt #q3)))
(y (/ (atan-qd/duplication arg) +pi+))
(true (/ #q6)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/dup.2
(let* ((arg (sqrt #q3))
(y (/ (atan-qd/duplication arg) +pi+))
(true (/ #q3)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/dup.3
(let* ((arg #q1)
(y (/ (atan-qd/duplication arg) +pi+))
(true (/ #q4)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/dup.4
(let* ((arg #q1q100)
(y (/ (atan-qd/duplication arg) +pi+))
(true #q.5))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/dup.5
(let* ((arg #q-1q100)
(y (/ (atan-qd/duplication arg) +pi+))
(true #q-.5))
(check-accuracy 212 y true))
nil)
;;; Tests of atan where we know the analytical result. Same tests,
;;; but using a CORDIC implementation.
(defun atan-qd/cordic (arg)
(make-instance 'qd-real
:value (octi::atan-qd/cordic (qd-value arg))))
(rt:deftest oct.atan/cordic.1
(let* ((arg (/ (sqrt #q3)))
(y (/ (atan-qd/cordic arg) +pi+))
(true (/ #q6)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/cordic.2
(let* ((arg (sqrt #q3))
(y (/ (atan-qd/cordic arg) +pi+))
(true (/ #q3)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/cordic.3
(let* ((arg #q1)
(y (/ (atan-qd/cordic arg) +pi+))
(true (/ #q4)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/cordic.4
(let* ((arg #q1q100)
(y (/ (atan-qd/cordic arg) +pi+))
(true #q.5))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.atan/cordic.5
(let* ((arg #q-1q100)
(y (/ (atan-qd/cordic arg) +pi+))
(true #q-.5))
(check-accuracy 212 y true))
nil)
;;; Tests of sin where we know the analytical result.
(rt:deftest oct.sin.1
(let* ((arg (/ +pi+ 6))
(y (sin arg))
(true #q.5))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sin.2
(let* ((arg (/ +pi+ 4))
(y (sin arg))
(true (sqrt #q.5)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sin.3
(let* ((arg (/ +pi+ 3))
(y (sin arg))
(true (/ (sqrt #q3) 2)))
(rt:deftest oct.big-sin.1
(let* ((arg (oct:make-qd (ash 1 120)))
(y (sin arg))
(true #q3.778201093607520226555484700569229919605866976512306642257987199414885q-1))
(check-accuracy 205 y true))
nil)
(rt:deftest oct.big-sin.2
(let* ((arg (oct:make-qd (ash 1 1023)))
(y (sin arg))
(true #q5.631277798508840134529434079444683477103854907361251399182750155357133q-1))
(check-accuracy 205 y true))
nil)
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;;; Tests of tan where we know the analytical result.
(rt:deftest oct.tan.1
(let* ((arg (/ +pi+ 6))
(y (tan arg))
(true (/ (sqrt #q3))))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.tan.2
(let* ((arg (/ +pi+ 4))
(y (tan arg))
(true #q1))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.tan.3
(let* ((arg (/ +pi+ 3))
(y (tan arg))
(true (sqrt #q3)))
(check-accuracy 212 y true))
nil)
;;; Tests of tan where we know the analytical result. Uses CORDIC
;;; algorithm.
(defun tan/cordic (arg)
(make-instance 'qd-real
:value (octi::tan-qd/cordic (qd-value arg))))
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(rt:deftest oct.tan/cordic.1
(let* ((arg (/ +pi+ 6))
(y (tan/cordic arg))
(true (/ (sqrt #q3))))
(check-accuracy 211 y true))
nil)
(rt:deftest oct.tan/cordic.2
(let* ((arg (/ +pi+ 4))
(y (tan/cordic arg))
(true #q1))
(check-accuracy 211 y true))
nil)
(rt:deftest oct.tan/cordic.3
(let* ((arg (/ +pi+ 3))
(y (tan/cordic arg))
(true (sqrt #q3)))
(check-accuracy 210 y true))
nil)
;;; Tests of asin where we know the analytical result.
(rt:deftest oct.asin.1
(let* ((arg #q.5)
(y (asin arg))
(true (/ +pi+ 6)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.asin.2
(let* ((arg (sqrt #q.5))
(y (asin arg))
(true (/ +pi+ 4)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.asin.3
(let* ((arg (/ (sqrt #q3) 2))
(y (asin arg))
(true (/ +pi+ 3)))
(check-accuracy 212 y true))
nil)
;;; Tests of log.
(rt:deftest oct.log.1
(let* ((arg #q2)
(y (log arg))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.log.2
(let* ((arg #q10)
(y (log arg))
(true (make-instance 'qd-real :value octi::+qd-log10+)))
(check-accuracy 207 y true))
nil)
(rt:deftest oct.log.3
(let* ((arg (+ 1 (scale-float #q1 -80)))
(y (log arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 212 y true))
nil)
;;; Tests of log using Newton iteration.
(defun log/newton (arg)
(make-instance 'qd-real
:value (octi::log-qd/newton (qd-value arg))))
(rt:deftest oct.log/newton.1
(let* ((arg #q2)
(y (log/newton arg))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.log/newton.2
(let* ((arg #q10)
(y (log/newton arg))
(true (make-instance 'qd-real :value octi::+qd-log10+)))
(check-accuracy 207 y true))
nil)
(rt:deftest oct.log/newton.3
(let* ((arg (+ 1 (scale-float #q1 -80)))
(y (log/newton arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 212 y true))
nil)
;;; Tests of log using AGM.
(defun log/agm (arg)
(make-instance 'qd-real
:value (octi::log-qd/agm (qd-value arg))))
(rt:deftest oct.log/agm.1
(let* ((arg #q2)
(y (log/agm arg))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
(check-accuracy 203 y true))
nil)
(rt:deftest oct.log/agm.2
(let* ((arg #q10)
(y (log/agm arg))
(true (make-instance 'qd-real :value octi::+qd-log10+)))
(check-accuracy 205 y true))
nil)
(rt:deftest oct.log/agm.3
(let* ((arg (+ 1 (scale-float #q1 -80)))
(y (log/agm arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 123 y true))
nil)
;;; Tests of log using AGM2, a faster variaton of AGM.
(defun log/agm2 (arg)
(make-instance 'qd-real
:value (octi::log-qd/agm2 (qd-value arg))))
(rt:deftest oct.log/agm2.1
(let* ((arg #q2)
(y (log/agm2 arg))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
(check-accuracy 203 y true))
nil)
(rt:deftest oct.log/agm2.2
(let* ((arg #q10)
(y (log/agm2 arg))
(true (make-instance 'qd-real :value octi::+qd-log10+)))
(check-accuracy 205 y true))
nil)
(rt:deftest oct.log/agm2.3
(let* ((arg (+ 1 (scale-float #q1 -80)))
(y (log/agm2 arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 123 y true))
nil)
;;; Tests of log using AGM3, a faster variation of AGM2.
(defun log/agm3 (arg)
(make-instance 'qd-real
:value (octi::log-qd/agm3 (qd-value arg))))
(rt:deftest oct.log/agm3.1
(let* ((arg #q2)
(y (log/agm3 arg))
(true (make-instance 'qd-real :value octi::+qd-log2+)))
(check-accuracy 203 y true))
nil)
(rt:deftest oct.log/agm3.2
(let* ((arg #q10)
(y (log/agm3 arg))
(true (make-instance 'qd-real :value octi::+qd-log10+)))
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(check-accuracy 205 y true))
nil)
(rt:deftest oct.log/agm3.3
(let* ((arg (+ 1 (scale-float #q1 -80)))
(y (log/agm3 arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 123 y true))
nil)
;;; Tests of sqrt to make sure we overflow or underflow where we
;;; shouldn't.
(rt:deftest oct.sqrt.1
(let* ((arg #q1q200)
(y (sqrt arg))
(true #q1q100))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sqrt.2
(let* ((arg #q1q200)
(y (sqrt arg))
(true #q1q100))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sqrt.3
(let* ((arg #q1q300)
(y (sqrt arg))
(true #q1q150))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sqrt.4
(let* ((arg #q1q-200)
(y (sqrt arg))
(true #q1q-100))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.sqrt.5
(let* ((arg #q1q-250)
(y (sqrt arg))
(true #q1q-125))
(check-accuracy 212 y true))
nil)
;;; Tests of log1p(x) = log(1+x), using the duplication formula.
(defun log1p/dup (arg)
(make-instance 'qd-real
:value (octi::log1p-qd/duplication (qd-value arg))))
(rt:deftest oct.log1p.1
(let* ((arg #q9)
(y (log1p/dup arg))
(true #q2.3025850929940456840179914546843642076011014886287729760333279009675726096773525q0))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.log1p.2
(let* ((arg (scale-float #q1 -80))
(y (log1p/dup arg))
(true #q8.2718061255302767487140834995607996176476940491239977084112840149578911975528492q-25))
(check-accuracy 212 y true))
nil)
;;; Tests of expm1(x) = exp(x) - 1, using a Taylor series with
;;; argument reduction.
(defun expm1/series (arg)
(make-instance 'qd-real
:value (octi::expm1-qd/series (qd-value arg))))
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(rt:deftest oct.expm1/series.1
(let* ((arg #q0)
(y (expm1/series arg))
(true #q0))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.expm1/series.2
(let* ((arg #q1)
(y (expm1/series arg))
(true #q1.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952q0))
(check-accuracy 211 y true))
nil)
(rt:deftest oct.expm1/series.3
(let* ((arg (scale-float #q1 -100))
(y (expm1/series arg))
(true #q7.888609052210118054117285652830973804370994921943802079729680186943164342372119432861876389514693341738324702996270767390039172777809233288470357147q-31))
(check-accuracy 211 y true))
nil)
;;; Tests of expm1(x) = exp(x) - 1, using duplication formula.
(defun expm1/dup (arg)
(make-instance 'qd-real
:value (octi::expm1-qd/duplication (qd-value arg))))
(rt:deftest oct.expm1/dup.1
(let* ((arg #q0)
(y (expm1/dup arg))
(true #q0))
(check-accuracy 212 y true))
nil)
(rt:deftest oct.expm1/dup.2
(let* ((arg #q1)
(y (expm1/dup arg))
(true #q1.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952q0))
(check-accuracy 211 y true))
nil)
(rt:deftest oct.expm1/dup.3
(let* ((arg (scale-float #q1 -100))
(y (expm1/dup arg))
(true #q7.888609052210118054117285652830973804370994921943802079729680186943164342372119432861876389514693341738324702996270767390039172777809233288470357147q-31))
(check-accuracy 211 y true))
nil)
;; If we screw up integer-decode-qd, printing is wrong. Here is one
;; case where integer-decode-qd was screwed up and printing the wrong
;; thing.
(rt:deftest oct.integer-decode.1
(multiple-value-bind (frac exp s)
(octi:integer-decode-qd (octi::%make-qd-d -0.03980126756814893d0
-2.7419792323327893d-18
0d0 0d0))
(unless (and (eql frac 103329998279901916046530991816704)
(eql exp -111)
(eql s -1))
(list frac exp s)))
nil)
;;;
;;; Add a few tests for the branch cuts. Many of these tests assume
;;; that Lisp has support for signed zeroes. If not, these tests are
;;; probably wrong.
(defun check-signs (fun arg expected)
(let* ((z (funcall fun arg))
(x (realpart z))
(y (imagpart z)))
;; If the Lisp doesn't support signed zeroes, then this test
;; should always pass.
(if (or (eql -0d0 0d0)
(and (= (float-sign x) (float-sign (realpart expected)))
(= (float-sign y) (float-sign (imagpart expected)))))
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
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735
736
737
738
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750
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781
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788
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800
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816
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819
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825
826
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829
t
(list z expected fun arg))))
;; asin has a branch cut on the real axis |x|>1. For x < -1, it is
;; continuous with quadrant II; for x > 1, continuous with quadrant
;; IV.
(rt:deftest oct.asin-branch-neg.1
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin -2d0 true))
t)
(rt:deftest oct.asin-branch-neg.2
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin #q-2 true))
t)
(rt:deftest oct.asin-branch-neg.3
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin #c(-2d0 0d0) true))
t)
(rt:deftest oct.asin-branch-neg.4
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin #q(-2 0) true))
t)
(rt:deftest oct.asin-branch-neg.5
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin #c(-2d0 -0d0) (conjugate true)))
t)
(rt:deftest oct.asin-branch-neg.6
(let ((true (cl:asin #c(-2d0 1d-20))))
(check-signs #'asin #q(-2d0 -0d0) (conjugate true)))
t)
(rt:deftest oct.asin-branch-pos.1
(let ((true (cl:asin #c(2d0 -1d-20))))
(check-signs #'asin #c(2d0 0d0) (conjugate true)))
t)
(rt:deftest oct.asin-branch-pos.2
(let ((true (cl:asin #c(2d0 -1d-20))))
(check-signs #'asin #q(2 0d0) (conjugate true)))
t)
(rt:deftest oct.asin-branch-pos.3
(let ((true (cl:asin #c(2d0 -1d-20))))
(check-signs #'asin #c(2d0 -0d0) true))
t)
(rt:deftest oct.asin-branch-pos.4
(let ((true (cl:asin #c(2d0 -1d-20))))
(check-signs #'asin #q(2d0 -0d0) true))
t)
;; acos branch cut is the real axis, |x| > 1. For x < -1, it is
;; continuous with quadrant II; for x > 1, quadrant IV.
(rt:deftest oct.acos-branch-neg.1
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos -2d0 true))
t)
(rt:deftest oct.acos-branch-neg.2
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos #q-2 true))
t)
(rt:deftest oct.acos-branch-neg.3
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos #c(-2d0 0d0) true))
t)
(rt:deftest oct.acos-branch-neg.4
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos #q(-2 0) true))
t)
(rt:deftest oct.acos-branch-neg.5
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos #c(-2d0 -0d0) (conjugate true)))
t)
(rt:deftest oct.acos-branch-neg.6
(let ((true (cl:acos #c(-2d0 1d-20))))
(check-signs #'acos #q(-2d0 -0d0) (conjugate true)))
t)
(rt:deftest oct.acos-branch-pos.1
(let ((true (cl:acos #c(2d0 -1d-20))))
(check-signs #'acos #c(2d0 0d0) (conjugate true)))
t)
(rt:deftest oct.acos-branch-pos.2
(let ((true (cl:acos #c(2d0 -1d-20))))
(check-signs #'acos #q(2 0d0) (conjugate true)))
t)
(rt:deftest oct.acos-branch-pos.3
(let ((true (cl:acos #c(2d0 -1d-20))))
(check-signs #'acos #c(2d0 -0d0) true))
t)
(rt:deftest oct.acos-branch-pos.4
(let ((true (cl:acos #c(2d0 -1d-20))))
(check-signs #'acos #q(2d0 -0d0) true))
t)
;; atan branch cut is the imaginary axis, |y| > 1. For y < -1, it is
;; continuous with quadrant IV; for x > 1, quadrant II.
(rt:deftest oct.atan-branch-neg.1
(let ((true (cl:atan #c(1d-20 -2d0))))
(check-signs #'atan #c(0d0 -2d0) true))
t)
(rt:deftest oct.atan-branch-neg.2
(let ((true (cl:atan #c(1d-20 -2d0))))
(check-signs #'atan #q(0 -2) true))
t)
(rt:deftest oct.atan-branch-neg.3
(let ((true (cl:atan #c(-1d-20 -2d0))))
(check-signs #'atan #c(-0d0 -2d0) true))
t)
(rt:deftest oct.atan-branch-neg.4
(let ((true (cl:atan #c(-1d-20 -2d0))))
(check-signs #'atan #q(-0d0 -2d0) true))
t)
(rt:deftest oct.atan-branch-pos.1
(let ((true (cl:atan #c(1d-20 2d0))))
(check-signs #'atan #c(0d0 2d0) true))
t)
(rt:deftest oct.atan-branch-pos.2
(let ((true (cl:atan #c(1d-20 2d0))))
t)
(rt:deftest oct.atan-branch-pos.3
(let ((true (cl:atan #c(-1d-20 2d0))))
(check-signs #'atan #c(-0d0 2d0) true))
t)
(rt:deftest oct.atan-branch-pos.4
(let ((true (cl:atan #c(-1d-20 2d0))))
(check-signs #'atan #q(-0d0 2d0) true))
t)
;; Test x < -1. CLHS says for x < -1, atanh is continuous with quadrant III.
(rt:deftest oct.atanh-branch-neg.1
(let ((true (cl:atanh #c(-2d0 -1d-20))))
(check-signs #'atanh -2d0 true))
t)
(rt:deftest oct.atanh-branch-neg.2
(let ((true (cl:atanh #c(-2d0 -1d-20))))
(check-signs #'atanh #q-2 true))
t)
;; Test x > 1. CLHS says for x > 1, atanh is continus with quadrant I.
(rt:deftest oct.atanh-branch-pos.1
(let ((true (cl:atanh #c(2d0 1d-20))))
(check-signs #'atanh 2d0 true))
t)
(rt:deftest oct.atanh-branch-pos.2
(let ((true (cl:atanh #c(2d0 1d-20))))
(check-signs #'atanh #q2 true))
t)
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
;; elliptic_k(-1) = gamma(1/4)^2/2^(5/2)/sqrt(%pi)
(rt:deftest oct.elliptic-k.1d
(let* ((val (elliptic-k -1d0))
(true #q1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547q0))
(check-accuracy 53 val true))
nil)
(rt:deftest oct.elliptic-k.1q
(let* ((val (elliptic-k #q-1q0))
(true #q1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547q0))
(check-accuracy 210 val true))
nil)
;; elliptic_k(1/2) = %pi^(3/2)/2/gamma(3/4)^2
(rt:deftest oct.elliptic-k.2d
(let* ((val (elliptic-k 0.5d0))
(true #q1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897q0))
(check-accuracy 53 val true))
nil)
(rt:deftest oct.elliptic-k.2q
(let* ((val (elliptic-k #q.5))
(true #q1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897q0))
(check-accuracy 210 val true))
nil)
;; jacobi_sn(K,1/2) = 1, where K = elliptic_k(1/2)
(rt:deftest oct.jacobi-sn.1d
(let* ((ek (elliptic-k .5d0))
(val (jacobi-sn ek .5d0)))
(check-accuracy 54 val 1d0))
nil)
(rt:deftest oct.jacobi-sn.1q
(let* ((ek (elliptic-k #q.5))
(val (jacobi-sn ek #q.5)))
(check-accuracy 212 val #q1))
nil)
;; jacobi_cn(K,1/2) = 0
(rt:deftest oct.jacobi-cn.1d
(let* ((ek (elliptic-k .5d0))
(val (jacobi-cn ek .5d0)))
(check-accuracy 50 val 0d0))
nil)
(let* ((ek (elliptic-k #q.5))
(val (jacobi-cn ek #q.5)))
(check-accuracy 210 val #q0))
nil)
;; jacobi-dn(K, 1/2) = sqrt(1/2)
(rt:deftest oct.jacobi-dn.1d
(let* ((ek (elliptic-k .5d0))
(true (sqrt .5d0))
(val (jacobi-dn ek .5d0)))
(check-accuracy 52 val true))
nil)
(rt:deftest oct.jacobi-dn.1q
(let* ((ek (elliptic-k #q.5))
(true (sqrt #q.5))
(val (jacobi-dn ek #q.5)))
(check-accuracy 212 val true))
nil)
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
(rt:deftest oct.carlson-rf.1d
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
;; = 1/4*beta(1/2,1/2)
;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4)
(let ((rf (carlson-rf 0d0 2d0 1d0))
(true 1.31102877714605990523241979494d0))
(check-accuracy 53 rf true))
nil)
(rt:deftest oct.carlson-rf.1q
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
(let ((rf (carlson-rf #q0 #q2 #q1))
(true #q1.311028777146059905232419794945559706841377475715811581408410851900395q0))
(check-accuracy 212 rf true))
nil)
(rt:deftest oct.carlson-rd.1d
;; Rd(0,2,1) = 3*integrate(s^2/sqrt(1-s^4), s, 0 ,1)
;; = 3*beta(3/4,1/2)/4
;; = 3*sqrt(%pi)*gamma(3/4)/gamma(1/4)
(let ((rd (carlson-rd 0d0 2d0 1d0))
(true 1.7972103521033883d0))
(check-accuracy 51 rd true))
nil)
(rt:deftest oct.carlson-rd.1q
(let ((rd (carlson-rd #q0 #q2 #q1))
(true #q1.797210352103388311159883738420485817340818994823477337395512429419599q0))
(check-accuracy 212 rd true))
nil)
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
;; Test some of the contagion stuff.
(rt:deftest oct.carlson-rf.contagion.1
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
;; = 1/4*beta(1/2,1/2)
;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4)
(let ((rf (carlson-rf 0 2 1))
(true 1.31102877714605990523241979494d0))
(check-accuracy 23 rf true))
nil)
(rt:deftest oct.carlson-rf.contagion.1d
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
;; = 1/4*beta(1/2,1/2)
;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4)
(let ((rf (carlson-rf 0d0 2 1))
(true 1.31102877714605990523241979494d0))
(check-accuracy 53 rf true))
nil)
(rt:deftest oct.carlson-rf.contagion.2d
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
;; = 1/4*beta(1/2,1/2)
;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4)
(let ((rf (carlson-rf 0 2d0 1))
(true 1.31102877714605990523241979494d0))
(check-accuracy 53 rf true))
nil)
(rt:deftest oct.carlson-rf.contagion.3d
;; Rf(0,2,1) = integrate(1/sqrt(1-s^4), s, 0 ,1)
;; = 1/4*beta(1/2,1/2)
;; = sqrt(%pi)/4*gamma(1/4)/gamma(3/4)
(let ((rf (carlson-rf 0 2 1d0))
(true 1.31102877714605990523241979494d0))
(check-accuracy 53 rf true))
nil)