Average Error: 57.9 → 0.8
Time: 36.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7216401 = 0.5;
        double r7216402 = re;
        double r7216403 = cos(r7216402);
        double r7216404 = r7216401 * r7216403;
        double r7216405 = 0.0;
        double r7216406 = im;
        double r7216407 = r7216405 - r7216406;
        double r7216408 = exp(r7216407);
        double r7216409 = exp(r7216406);
        double r7216410 = r7216408 - r7216409;
        double r7216411 = r7216404 * r7216410;
        return r7216411;
}
↓
double f(double re, double im) {
        double r7216412 = -0.016666666666666666;
        double r7216413 = im;
        double r7216414 = 5.0;
        double r7216415 = pow(r7216413, r7216414);
        double r7216416 = r7216412 * r7216415;
        double r7216417 = r7216413 + r7216413;
        double r7216418 = r7216416 - r7216417;
        double r7216419 = r7216413 * r7216413;
        double r7216420 = 0.3333333333333333;
        double r7216421 = r7216413 * r7216420;
        double r7216422 = r7216419 * r7216421;
        double r7216423 = r7216418 - r7216422;
        double r7216424 = 0.5;
        double r7216425 = re;
        double r7216426 = cos(r7216425);
        double r7216427 = r7216424 * r7216426;
        double r7216428 = r7216423 * r7216427;
        return r7216428;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019162 
(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))))