Average Error: 58.2 → 0.7
Time: 32.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\cos re \cdot 0.5\right) \cdot \left(\left(-\left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\cos re \cdot 0.5\right) \cdot \left(\left(-\left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)
double f(double re, double im) {
        double r7229793 = 0.5;
        double r7229794 = re;
        double r7229795 = cos(r7229794);
        double r7229796 = r7229793 * r7229795;
        double r7229797 = 0.0;
        double r7229798 = im;
        double r7229799 = r7229797 - r7229798;
        double r7229800 = exp(r7229799);
        double r7229801 = exp(r7229798);
        double r7229802 = r7229800 - r7229801;
        double r7229803 = r7229796 * r7229802;
        return r7229803;
}
↓
double f(double re, double im) {
        double r7229804 = re;
        double r7229805 = cos(r7229804);
        double r7229806 = 0.5;
        double r7229807 = r7229805 * r7229806;
        double r7229808 = im;
        double r7229809 = r7229808 + r7229808;
        double r7229810 = -r7229809;
        double r7229811 = 0.016666666666666666;
        double r7229812 = 5.0;
        double r7229813 = pow(r7229808, r7229812);
        double r7229814 = r7229811 * r7229813;
        double r7229815 = r7229810 - r7229814;
        double r7229816 = r7229807 * r7229815;
        double r7229817 = r7229808 * r7229808;
        double r7229818 = r7229808 * r7229817;
        double r7229819 = -0.3333333333333333;
        double r7229820 = r7229818 * r7229819;
        double r7229821 = r7229820 * r7229807;
        double r7229822 = r7229816 + r7229821;
        return r7229822;
}

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

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

  5. Applied sub-neg0.7

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

  6. Applied associate--l+0.7

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

  7. Applied distribute-rgt-in0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 
(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))))