Average Error: 43.8 → 0.7
Time: 36.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \mathsf{fma}\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right), \frac{-1}{60}, -2 \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \mathsf{fma}\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right), \frac{-1}{60}, -2 \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r7826439 = 0.5;
        double r7826440 = re;
        double r7826441 = sin(r7826440);
        double r7826442 = r7826439 * r7826441;
        double r7826443 = im;
        double r7826444 = -r7826443;
        double r7826445 = exp(r7826444);
        double r7826446 = exp(r7826443);
        double r7826447 = r7826445 - r7826446;
        double r7826448 = r7826442 * r7826447;
        return r7826448;
}
↓
double f(double re, double im) {
        double r7826449 = im;
        double r7826450 = r7826449 * r7826449;
        double r7826451 = r7826449 * r7826450;
        double r7826452 = -0.3333333333333333;
        double r7826453 = r7826451 * r7826452;
        double r7826454 = 0.5;
        double r7826455 = re;
        double r7826456 = sin(r7826455);
        double r7826457 = r7826454 * r7826456;
        double r7826458 = r7826453 * r7826457;
        double r7826459 = r7826451 * r7826450;
        double r7826460 = -0.016666666666666666;
        double r7826461 = -2.0;
        double r7826462 = r7826461 * r7826449;
        double r7826463 = fma(r7826459, r7826460, r7826462);
        double r7826464 = r7826463 * r7826457;
        double r7826465 = r7826458 + r7826464;
        return r7826465;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.8
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.8

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

  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. Simplified0.7

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))