Average Error: 58.1 → 0.7
Time: 35.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r9358501 = 0.5;
        double r9358502 = re;
        double r9358503 = cos(r9358502);
        double r9358504 = r9358501 * r9358503;
        double r9358505 = 0.0;
        double r9358506 = im;
        double r9358507 = r9358505 - r9358506;
        double r9358508 = exp(r9358507);
        double r9358509 = exp(r9358506);
        double r9358510 = r9358508 - r9358509;
        double r9358511 = r9358504 * r9358510;
        return r9358511;
}
↓
double f(double re, double im) {
        double r9358512 = -0.3333333333333333;
        double r9358513 = im;
        double r9358514 = r9358513 * r9358513;
        double r9358515 = r9358513 * r9358514;
        double r9358516 = r9358512 * r9358515;
        double r9358517 = 5.0;
        double r9358518 = pow(r9358513, r9358517);
        double r9358519 = -0.016666666666666666;
        double r9358520 = r9358518 * r9358519;
        double r9358521 = r9358513 + r9358513;
        double r9358522 = r9358520 - r9358521;
        double r9358523 = r9358516 + r9358522;
        double r9358524 = 0.5;
        double r9358525 = re;
        double r9358526 = cos(r9358525);
        double r9358527 = r9358524 * r9358526;
        double r9358528 = r9358523 * r9358527;
        return r9358528;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))