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improve accuracy of tanh function (#2565)

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improve accuracy of tanh function

using the implementation of https://www.math.utah.edu/~beebe/software/ieee/tanh.pdf

by changing from (/ (- 1 exp2z) (+ 1 exp2z)) to (- 1 (/ 2 (+ 1 exp2z)) the accuracy after rounding is increased (I was comparing with bftanh) and removes the fluctuations around z=18.35
using the polynomial for z ϵ(1.290e-8 to 0.549) seems to increase the accuracy after rounding even further
see comparison: http://pasterack.org/pastes/48436

especially the fact that  (< (tanh 18.36)(tanh 18.37)) ;=> #t was tripping me up
the two extra conditions (z . < . 1.29e-8) and (z . < . 0.549) are optional to solve this
This commit is contained in:
bdeket 2019-03-28 12:36:18 +01:00 committed by Matthew Flatt
parent 82d5b46819
commit eea771ea64
1 changed files with 17 additions and 6 deletions
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23
racket/collects/racket/math.rkt
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@ -65,18 +65,29 @@
[else (/ (+ (exp z) (exp (- z))) 2)]))
(define (tanh z)
;implementation based on https://www.math.utah.edu/~beebe/software/ieee/tanh.pdf
(unless (number? z) (raise-argument-error 'tanh "number?" z))
(cond [(= z 0) z] ; preserve 0, 0.0, -0.0, 0.0f0, 0.0+0.0i, etc.
[(real? z)
(let loop ([z z])
(cond [(z . < . 0) (- (loop (- z)))]
[(z . < . 20) (define exp2z (exp (* 2 z)))
(/ (- exp2z 1) (+ exp2z 1))]
[(z . >= . 20) (if (single-flonum? z) 1.0f0 1.0)]
[(z . < . 1.29047841397589243466D-08) z]
[(z . < . 0.54930614433405484570D+00)
(define p0 -0.16134119023996228053D+04)
(define p1 -0.99225929672236083313D+02)
(define p2 -0.96437492777225469787D+00)
(define q0 0.48402357071988688686D+04)
(define q1 0.22337720718962312926D+04)
(define q2 0.11274474380534949335D+03)
(define g (* z z))
(define R
(/ (* g (+ (* (+ (* p2 g) p1) g) p0))
(+ (* (+ (* (+ g q2) g) q1) g) q0)))
(+ z (* z R))]
[(z . < . 19.06154746539849600897D+00) (- 1 (/ 2 (+ 1 (exp (* 2 z)))))]
[(z . >= . 19.06154746539849600897D+00) (if (single-flonum? z) 1.0f0 1.0)]
[else z]))] ; +nan.0 or +nan.f
[else
(define exp2z (exp (* 2 z)))
(/ (- exp2z 1) (+ exp2z 1))]))
[else (- 1 (/ 2 (+ 1 (exp (* 2 z)))))]))
;; angle conversion
(define (degrees->radians x)