Average Error: 58.3 → 0.6
Time: 35.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r6922352 = 0.5;
        double r6922353 = re;
        double r6922354 = cos(r6922353);
        double r6922355 = r6922352 * r6922354;
        double r6922356 = 0.0;
        double r6922357 = im;
        double r6922358 = r6922356 - r6922357;
        double r6922359 = exp(r6922358);
        double r6922360 = exp(r6922357);
        double r6922361 = r6922359 - r6922360;
        double r6922362 = r6922355 * r6922361;
        return r6922362;
}
↓
double f(double re, double im) {
        double r6922363 = im;
        double r6922364 = r6922363 * r6922363;
        double r6922365 = r6922363 * r6922364;
        double r6922366 = -0.3333333333333333;
        double r6922367 = r6922365 * r6922366;
        double r6922368 = 0.016666666666666666;
        double r6922369 = 5.0;
        double r6922370 = pow(r6922363, r6922369);
        double r6922371 = r6922363 + r6922363;
        double r6922372 = fma(r6922368, r6922370, r6922371);
        double r6922373 = r6922367 - r6922372;
        double r6922374 = 0.5;
        double r6922375 = re;
        double r6922376 = cos(r6922375);
        double r6922377 = r6922374 * r6922376;
        double r6922378 = r6922373 * r6922377;
        return r6922378;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.3
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.3

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))