Average Error: 43.7 → 0.8
Time: 24.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} - {im}^{5} \cdot \frac{1}{60}\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} - {im}^{5} \cdot \frac{1}{60}\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r143735 = 0.5;
        double r143736 = re;
        double r143737 = sin(r143736);
        double r143738 = r143735 * r143737;
        double r143739 = im;
        double r143740 = -r143739;
        double r143741 = exp(r143740);
        double r143742 = exp(r143739);
        double r143743 = r143741 - r143742;
        double r143744 = r143738 * r143743;
        return r143744;
}
↓
double f(double re, double im) {
        double r143745 = im;
        double r143746 = -2.0;
        double r143747 = r143745 * r143746;
        double r143748 = 3.0;
        double r143749 = pow(r143745, r143748);
        double r143750 = -0.3333333333333333;
        double r143751 = r143749 * r143750;
        double r143752 = 5.0;
        double r143753 = pow(r143745, r143752);
        double r143754 = 0.016666666666666666;
        double r143755 = r143753 * r143754;
        double r143756 = r143751 - r143755;
        double r143757 = r143747 + r143756;
        double r143758 = 0.5;
        double r143759 = re;
        double r143760 = sin(r143759);
        double r143761 = r143758 * r143760;
        double r143762 = r143757 * r143761;
        return r143762;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied pow10.8

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

  6. Applied pow10.8

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

  7. Applied pow10.8

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

  8. Applied pow-prod-down0.8

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

  9. Applied pow-prod-down0.8

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

  10. Simplified0.8

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

  11. Final simplification0.8

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

Reproduce

herbie shell --seed 2019194 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))