Average Error: 43.3 → 0.9
Time: 30.0s
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 r3978047 = 0.5;
        double r3978048 = re;
        double r3978049 = sin(r3978048);
        double r3978050 = r3978047 * r3978049;
        double r3978051 = im;
        double r3978052 = -r3978051;
        double r3978053 = exp(r3978052);
        double r3978054 = exp(r3978051);
        double r3978055 = r3978053 - r3978054;
        double r3978056 = r3978050 * r3978055;
        return r3978056;
}
↓
double f(double re, double im) {
        double r3978057 = -0.3333333333333333;
        double r3978058 = im;
        double r3978059 = r3978058 * r3978058;
        double r3978060 = r3978058 * r3978059;
        double r3978061 = r3978057 * r3978060;
        double r3978062 = 0.016666666666666666;
        double r3978063 = 5.0;
        double r3978064 = pow(r3978058, r3978063);
        double r3978065 = r3978058 + r3978058;
        double r3978066 = fma(r3978062, r3978064, r3978065);
        double r3978067 = r3978061 - r3978066;
        double r3978068 = 0.5;
        double r3978069 = re;
        double r3978070 = sin(r3978069);
        double r3978071 = r3978068 * r3978070;
        double r3978072 = r3978067 * r3978071;
        return r3978072;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.3
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 43.3

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

  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

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019154 +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))))