Average Error: 57.9 → 0.8
Time: 11.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)
double f(double re, double im) {
        double r170815 = 0.5;
        double r170816 = re;
        double r170817 = cos(r170816);
        double r170818 = r170815 * r170817;
        double r170819 = 0.0;
        double r170820 = im;
        double r170821 = r170819 - r170820;
        double r170822 = exp(r170821);
        double r170823 = exp(r170820);
        double r170824 = r170822 - r170823;
        double r170825 = r170818 * r170824;
        return r170825;
}
↓
double f(double re, double im) {
        double r170826 = 0.5;
        double r170827 = re;
        double r170828 = cos(r170827);
        double r170829 = r170826 * r170828;
        double r170830 = 0.3333333333333333;
        double r170831 = im;
        double r170832 = 3.0;
        double r170833 = pow(r170831, r170832);
        double r170834 = r170830 * r170833;
        double r170835 = -r170834;
        double r170836 = 0.016666666666666666;
        double r170837 = 5.0;
        double r170838 = pow(r170831, r170837);
        double r170839 = 2.0;
        double r170840 = r170839 * r170831;
        double r170841 = fma(r170836, r170838, r170840);
        double r170842 = r170835 - r170841;
        double r170843 = r170829 * r170842;
        return r170843;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.9
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))