Average Error: 43.5 → 0.9
Time: 42.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right) + \mathsf{fma}\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right), \frac{-1}{60}, im \cdot -2\right) \cdot \left(0.5 \cdot \sin re\right)\]
\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)\right) \cdot \left(0.5 \cdot \sin re\right) + \mathsf{fma}\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right), \frac{-1}{60}, im \cdot -2\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r10589975 = 0.5;
        double r10589976 = re;
        double r10589977 = sin(r10589976);
        double r10589978 = r10589975 * r10589977;
        double r10589979 = im;
        double r10589980 = -r10589979;
        double r10589981 = exp(r10589980);
        double r10589982 = exp(r10589979);
        double r10589983 = r10589981 - r10589982;
        double r10589984 = r10589978 * r10589983;
        return r10589984;
}
↓
double f(double re, double im) {
        double r10589985 = -0.3333333333333333;
        double r10589986 = im;
        double r10589987 = r10589986 * r10589986;
        double r10589988 = r10589986 * r10589987;
        double r10589989 = r10589985 * r10589988;
        double r10589990 = 0.5;
        double r10589991 = re;
        double r10589992 = sin(r10589991);
        double r10589993 = r10589990 * r10589992;
        double r10589994 = r10589989 * r10589993;
        double r10589995 = r10589987 * r10589987;
        double r10589996 = r10589986 * r10589995;
        double r10589997 = -0.016666666666666666;
        double r10589998 = -2.0;
        double r10589999 = r10589986 * r10589998;
        double r10590000 = fma(r10589996, r10589997, r10589999);
        double r10590001 = r10590000 * r10589993;
        double r10590002 = r10589994 + r10590001;
        return r10590002;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

  3. Simplified0.9

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

  5. Applied sub-neg0.9

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

  6. Applied distribute-lft-in0.9

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

  7. Simplified0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2019143 +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))))