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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+)))
;;; 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)))
(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
;; We know atan(10^100) is pi/2 because atan(10^100) =
;; pi/2-atan(10^-100) and atan(10^-100) is approx 10^-100, which
;; is too small to affect pi/2.
(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)))))
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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)
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;; 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)
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(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)
;; 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)