Average Error: 43.4 → 0.7
Time: 42.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left(0.5 \cdot \sin re\right) \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} + -2\right)\]
double f(double re, double im) {
        double r45642880 = 0.5;
        double r45642881 = re;
        double r45642882 = sin(r45642881);
        double r45642883 = r45642880 * r45642882;
        double r45642884 = im;
        double r45642885 = -r45642884;
        double r45642886 = exp(r45642885);
        double r45642887 = exp(r45642884);
        double r45642888 = r45642886 - r45642887;
        double r45642889 = r45642883 * r45642888;
        return r45642889;
}
↓
double f(double re, double im) {
        double r45642890 = im;
        double r45642891 = 5.0;
        double r45642892 = pow(r45642890, r45642891);
        double r45642893 = -0.016666666666666666;
        double r45642894 = r45642892 * r45642893;
        double r45642895 = 0.5;
        double r45642896 = re;
        double r45642897 = sin(r45642896);
        double r45642898 = r45642895 * r45642897;
        double r45642899 = r45642894 * r45642898;
        double r45642900 = r45642898 * r45642890;
        double r45642901 = r45642890 * r45642890;
        double r45642902 = -0.3333333333333333;
        double r45642903 = r45642901 * r45642902;
        double r45642904 = -2.0;
        double r45642905 = r45642903 + r45642904;
        double r45642906 = r45642900 * r45642905;
        double r45642907 = r45642899 + r45642906;
        return r45642907;
}


\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left(0.5 \cdot \sin re\right) \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{3} + -2\right)

Error

Bits error versus re

Bits error versus im

Target

Original43.4
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 43.4

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

  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

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019102 
(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))))