Average Error: 57.9 → 0.8
Time: 34.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), {im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), {im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r6810882 = 0.5;
        double r6810883 = re;
        double r6810884 = cos(r6810883);
        double r6810885 = r6810882 * r6810884;
        double r6810886 = 0.0;
        double r6810887 = im;
        double r6810888 = r6810886 - r6810887;
        double r6810889 = exp(r6810888);
        double r6810890 = exp(r6810887);
        double r6810891 = r6810889 - r6810890;
        double r6810892 = r6810885 * r6810891;
        return r6810892;
}
↓
double f(double re, double im) {
        double r6810893 = -0.3333333333333333;
        double r6810894 = im;
        double r6810895 = r6810894 * r6810894;
        double r6810896 = r6810894 * r6810895;
        double r6810897 = 5.0;
        double r6810898 = pow(r6810894, r6810897);
        double r6810899 = -0.016666666666666666;
        double r6810900 = r6810898 * r6810899;
        double r6810901 = r6810894 + r6810894;
        double r6810902 = r6810900 - r6810901;
        double r6810903 = fma(r6810893, r6810896, r6810902);
        double r6810904 = 0.5;
        double r6810905 = re;
        double r6810906 = cos(r6810905);
        double r6810907 = r6810904 * r6810906;
        double r6810908 = r6810903 * r6810907;
        return r6810908;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.9
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))