Average Error: 58.0 → 0.8
Time: 37.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r8146601 = 0.5;
        double r8146602 = re;
        double r8146603 = cos(r8146602);
        double r8146604 = r8146601 * r8146603;
        double r8146605 = 0.0;
        double r8146606 = im;
        double r8146607 = r8146605 - r8146606;
        double r8146608 = exp(r8146607);
        double r8146609 = exp(r8146606);
        double r8146610 = r8146608 - r8146609;
        double r8146611 = r8146604 * r8146610;
        return r8146611;
}
↓
double f(double re, double im) {
        double r8146612 = -0.3333333333333333;
        double r8146613 = im;
        double r8146614 = r8146613 * r8146613;
        double r8146615 = r8146613 * r8146614;
        double r8146616 = r8146612 * r8146615;
        double r8146617 = r8146613 + r8146613;
        double r8146618 = r8146616 - r8146617;
        double r8146619 = 0.016666666666666666;
        double r8146620 = 5.0;
        double r8146621 = pow(r8146613, r8146620);
        double r8146622 = r8146619 * r8146621;
        double r8146623 = r8146618 - r8146622;
        double r8146624 = 0.5;
        double r8146625 = re;
        double r8146626 = cos(r8146625);
        double r8146627 = r8146624 * r8146626;
        double r8146628 = r8146623 * r8146627;
        return r8146628;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019179 
(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))))