Average Error: 58.2 → 0.7
Time: 36.7s
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 r7185871 = 0.5;
        double r7185872 = re;
        double r7185873 = cos(r7185872);
        double r7185874 = r7185871 * r7185873;
        double r7185875 = 0.0;
        double r7185876 = im;
        double r7185877 = r7185875 - r7185876;
        double r7185878 = exp(r7185877);
        double r7185879 = exp(r7185876);
        double r7185880 = r7185878 - r7185879;
        double r7185881 = r7185874 * r7185880;
        return r7185881;
}
↓
double f(double re, double im) {
        double r7185882 = -0.3333333333333333;
        double r7185883 = im;
        double r7185884 = r7185883 * r7185883;
        double r7185885 = r7185883 * r7185884;
        double r7185886 = 5.0;
        double r7185887 = pow(r7185883, r7185886);
        double r7185888 = -0.016666666666666666;
        double r7185889 = r7185887 * r7185888;
        double r7185890 = r7185883 + r7185883;
        double r7185891 = r7185889 - r7185890;
        double r7185892 = fma(r7185882, r7185885, r7185891);
        double r7185893 = 0.5;
        double r7185894 = re;
        double r7185895 = cos(r7185894);
        double r7185896 = r7185893 * r7185895;
        double r7185897 = r7185892 * r7185896;
        return r7185897;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.2
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 +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))))