Average Error: 43.7 → 0.9
Time: 1.1m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + \left({im}^{5} \cdot \frac{1}{60} + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + \left({im}^{5} \cdot \frac{1}{60} + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r7208822 = 0.5;
        double r7208823 = re;
        double r7208824 = sin(r7208823);
        double r7208825 = r7208822 * r7208824;
        double r7208826 = im;
        double r7208827 = -r7208826;
        double r7208828 = exp(r7208827);
        double r7208829 = exp(r7208826);
        double r7208830 = r7208828 - r7208829;
        double r7208831 = r7208825 * r7208830;
        return r7208831;
}
↓
double f(double re, double im) {
        double r7208832 = -0.3333333333333333;
        double r7208833 = im;
        double r7208834 = r7208833 * r7208833;
        double r7208835 = r7208833 * r7208834;
        double r7208836 = r7208832 * r7208835;
        double r7208837 = 5.0;
        double r7208838 = pow(r7208833, r7208837);
        double r7208839 = 0.016666666666666666;
        double r7208840 = r7208838 * r7208839;
        double r7208841 = r7208840 + r7208833;
        double r7208842 = r7208833 + r7208841;
        double r7208843 = r7208836 - r7208842;
        double r7208844 = 0.5;
        double r7208845 = re;
        double r7208846 = sin(r7208845);
        double r7208847 = r7208844 * r7208846;
        double r7208848 = r7208843 * r7208847;
        return r7208848;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1.0:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019165 
(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))))