math.cos on complex, imaginary part

Average Error: 43.2 → 0.7

Time: 32.3s

Precision: 64

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

↓

\[\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3}\right) + \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\]

C

double f(double re, double im) {
        double r137332 = 0.5;
        double r137333 = re;
        double r137334 = sin(r137333);
        double r137335 = r137332 * r137334;
        double r137336 = im;
        double r137337 = -r137336;
        double r137338 = exp(r137337);
        double r137339 = exp(r137336);
        double r137340 = r137338 - r137339;
        double r137341 = r137335 * r137340;
        return r137341;
}

↓

double f(double re, double im) {
        double r137342 = 0.5;
        double r137343 = re;
        double r137344 = sin(r137343);
        double r137345 = r137342 * r137344;
        double r137346 = -0.3333333333333333;
        double r137347 = im;
        double r137348 = 3.0;
        double r137349 = pow(r137347, r137348);
        double r137350 = r137346 * r137349;
        double r137351 = r137345 * r137350;
        double r137352 = -2.0;
        double r137353 = 5.0;
        double r137354 = pow(r137347, r137353);
        double r137355 = -0.016666666666666666;
        double r137356 = r137354 * r137355;
        double r137357 = fma(r137347, r137352, r137356);
        double r137358 = r137345 * r137357;
        double r137359 = r137351 + r137358;
        return r137359;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.2
Target	0.3
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.2

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

2. Taylor expanded around 0 0.7

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

3. Simplified 0.7

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {im}^{3}, \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)}\]
4. Using strategy rm

5. Applied fma-udef 0.7

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{3} + \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)}\]

6. Applied distribute-lft-in 0.7

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3}\right) + \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)}\]

7. Final simplification 0.7

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3}\right) + \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))