Average Error: 58.2 → 0.7
Time: 1.8m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r24280175 = 0.5;
        double r24280176 = re;
        double r24280177 = cos(r24280176);
        double r24280178 = r24280175 * r24280177;
        double r24280179 = 0.0;
        double r24280180 = im;
        double r24280181 = r24280179 - r24280180;
        double r24280182 = exp(r24280181);
        double r24280183 = exp(r24280180);
        double r24280184 = r24280182 - r24280183;
        double r24280185 = r24280178 * r24280184;
        return r24280185;
}
↓
double f(double re, double im) {
        double r24280186 = im;
        double r24280187 = 5.0;
        double r24280188 = pow(r24280186, r24280187);
        double r24280189 = -0.016666666666666666;
        double r24280190 = -0.3333333333333333;
        double r24280191 = r24280186 * r24280190;
        double r24280192 = r24280186 * r24280191;
        double r24280193 = 2.0;
        double r24280194 = r24280192 - r24280193;
        double r24280195 = r24280186 * r24280194;
        double r24280196 = fma(r24280188, r24280189, r24280195);
        double r24280197 = 0.5;
        double r24280198 = re;
        double r24280199 = cos(r24280198);
        double r24280200 = r24280197 * r24280199;
        double r24280201 = r24280196 * r24280200;
        return r24280201;
}

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 0.6

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

  3. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]

Reproduce

herbie shell --seed 2019121 +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))))