Average Error: 43.5 → 0.7
Time: 35.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, \mathsf{fma}\left(\frac{1}{3}, im \cdot im, 2\right) \cdot \left(-im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, \mathsf{fma}\left(\frac{1}{3}, im \cdot im, 2\right) \cdot \left(-im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r8354737 = 0.5;
        double r8354738 = re;
        double r8354739 = sin(r8354738);
        double r8354740 = r8354737 * r8354739;
        double r8354741 = im;
        double r8354742 = -r8354741;
        double r8354743 = exp(r8354742);
        double r8354744 = exp(r8354741);
        double r8354745 = r8354743 - r8354744;
        double r8354746 = r8354740 * r8354745;
        return r8354746;
}
↓
double f(double re, double im) {
        double r8354747 = -0.016666666666666666;
        double r8354748 = im;
        double r8354749 = 5.0;
        double r8354750 = pow(r8354748, r8354749);
        double r8354751 = 0.3333333333333333;
        double r8354752 = r8354748 * r8354748;
        double r8354753 = 2.0;
        double r8354754 = fma(r8354751, r8354752, r8354753);
        double r8354755 = -r8354748;
        double r8354756 = r8354754 * r8354755;
        double r8354757 = fma(r8354747, r8354750, r8354756);
        double r8354758 = 0.5;
        double r8354759 = re;
        double r8354760 = sin(r8354759);
        double r8354761 = r8354758 * r8354760;
        double r8354762 = r8354757 * r8354761;
        return r8354762;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

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