Average Error: 43.8 → 0.8
Time: 10.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r266766 = 0.5;
        double r266767 = re;
        double r266768 = sin(r266767);
        double r266769 = r266766 * r266768;
        double r266770 = im;
        double r266771 = -r266770;
        double r266772 = exp(r266771);
        double r266773 = exp(r266770);
        double r266774 = r266772 - r266773;
        double r266775 = r266769 * r266774;
        return r266775;
}
↓
double f(double re, double im) {
        double r266776 = 0.5;
        double r266777 = re;
        double r266778 = sin(r266777);
        double r266779 = r266776 * r266778;
        double r266780 = 0.3333333333333333;
        double r266781 = im;
        double r266782 = 3.0;
        double r266783 = pow(r266781, r266782);
        double r266784 = r266780 * r266783;
        double r266785 = 0.016666666666666666;
        double r266786 = 5.0;
        double r266787 = pow(r266781, r266786);
        double r266788 = r266785 * r266787;
        double r266789 = 2.0;
        double r266790 = r266789 * r266781;
        double r266791 = r266788 + r266790;
        double r266792 = r266784 + r266791;
        double r266793 = -r266792;
        double r266794 = r266779 * r266793;
        return r266794;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.8
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.8

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

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020024 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))