Average Error: 43.3 → 0.9
Time: 27.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r5458284 = 0.5;
        double r5458285 = re;
        double r5458286 = sin(r5458285);
        double r5458287 = r5458284 * r5458286;
        double r5458288 = im;
        double r5458289 = -r5458288;
        double r5458290 = exp(r5458289);
        double r5458291 = exp(r5458288);
        double r5458292 = r5458290 - r5458291;
        double r5458293 = r5458287 * r5458292;
        return r5458293;
}
↓
double f(double re, double im) {
        double r5458294 = im;
        double r5458295 = r5458294 * r5458294;
        double r5458296 = r5458294 * r5458295;
        double r5458297 = -0.3333333333333333;
        double r5458298 = r5458296 * r5458297;
        double r5458299 = 5.0;
        double r5458300 = pow(r5458294, r5458299);
        double r5458301 = 0.016666666666666666;
        double r5458302 = r5458300 * r5458301;
        double r5458303 = r5458294 + r5458294;
        double r5458304 = r5458302 + r5458303;
        double r5458305 = r5458298 - r5458304;
        double r5458306 = 0.5;
        double r5458307 = re;
        double r5458308 = sin(r5458307);
        double r5458309 = r5458306 * r5458308;
        double r5458310 = r5458305 * r5458309;
        return r5458310;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019154 
(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))))