Average Error: 43.1 → 0.8
Time: 29.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r3826876 = 0.5;
        double r3826877 = re;
        double r3826878 = sin(r3826877);
        double r3826879 = r3826876 * r3826878;
        double r3826880 = im;
        double r3826881 = -r3826880;
        double r3826882 = exp(r3826881);
        double r3826883 = exp(r3826880);
        double r3826884 = r3826882 - r3826883;
        double r3826885 = r3826879 * r3826884;
        return r3826885;
}
↓
double f(double re, double im) {
        double r3826886 = -0.3333333333333333;
        double r3826887 = im;
        double r3826888 = r3826887 * r3826887;
        double r3826889 = r3826887 * r3826888;
        double r3826890 = r3826886 * r3826889;
        double r3826891 = 0.016666666666666666;
        double r3826892 = 5.0;
        double r3826893 = pow(r3826887, r3826892);
        double r3826894 = r3826887 + r3826887;
        double r3826895 = fma(r3826891, r3826893, r3826894);
        double r3826896 = r3826890 - r3826895;
        double r3826897 = 0.5;
        double r3826898 = re;
        double r3826899 = sin(r3826898);
        double r3826900 = r3826897 * r3826899;
        double r3826901 = r3826896 * r3826900;
        return r3826901;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.1
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.1

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019156 +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))))