Average Error: 58.2 → 0.7
Time: 26.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{3} - \left(im + im\right)\right) - \left(\sqrt{\frac{1}{60}} \cdot {im}^{5}\right) \cdot \sqrt{\frac{1}{60}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{3} - \left(im + im\right)\right) - \left(\sqrt{\frac{1}{60}} \cdot {im}^{5}\right) \cdot \sqrt{\frac{1}{60}}\right)
double f(double re, double im) {
        double r161613 = 0.5;
        double r161614 = re;
        double r161615 = cos(r161614);
        double r161616 = r161613 * r161615;
        double r161617 = 0.0;
        double r161618 = im;
        double r161619 = r161617 - r161618;
        double r161620 = exp(r161619);
        double r161621 = exp(r161618);
        double r161622 = r161620 - r161621;
        double r161623 = r161616 * r161622;
        return r161623;
}
↓
double f(double re, double im) {
        double r161624 = 0.5;
        double r161625 = re;
        double r161626 = cos(r161625);
        double r161627 = r161624 * r161626;
        double r161628 = -0.3333333333333333;
        double r161629 = im;
        double r161630 = 3.0;
        double r161631 = pow(r161629, r161630);
        double r161632 = r161628 * r161631;
        double r161633 = r161629 + r161629;
        double r161634 = r161632 - r161633;
        double r161635 = 0.016666666666666666;
        double r161636 = sqrt(r161635);
        double r161637 = 5.0;
        double r161638 = pow(r161629, r161637);
        double r161639 = r161636 * r161638;
        double r161640 = r161639 * r161636;
        double r161641 = r161634 - r161640;
        double r161642 = r161627 * r161641;
        return r161642;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
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 58.2

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

  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

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

  5. Applied add-sqr-sqrt0.7

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

  6. Applied associate-*l*0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019196 
(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))))