Average Error: 43.5 → 0.9
Time: 32.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot \log \left(e^{im \cdot \left(im \cdot im\right)}\right) - \left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot \log \left(e^{im \cdot \left(im \cdot im\right)}\right) - \left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right)\right)
double f(double re, double im) {
        double r8367842 = 0.5;
        double r8367843 = re;
        double r8367844 = sin(r8367843);
        double r8367845 = r8367842 * r8367844;
        double r8367846 = im;
        double r8367847 = -r8367846;
        double r8367848 = exp(r8367847);
        double r8367849 = exp(r8367846);
        double r8367850 = r8367848 - r8367849;
        double r8367851 = r8367845 * r8367850;
        return r8367851;
}
↓
double f(double re, double im) {
        double r8367852 = 0.5;
        double r8367853 = re;
        double r8367854 = sin(r8367853);
        double r8367855 = r8367852 * r8367854;
        double r8367856 = -0.3333333333333333;
        double r8367857 = im;
        double r8367858 = r8367857 * r8367857;
        double r8367859 = r8367857 * r8367858;
        double r8367860 = exp(r8367859);
        double r8367861 = log(r8367860);
        double r8367862 = r8367856 * r8367861;
        double r8367863 = r8367857 + r8367857;
        double r8367864 = 0.016666666666666666;
        double r8367865 = 5.0;
        double r8367866 = pow(r8367857, r8367865);
        double r8367867 = r8367864 * r8367866;
        double r8367868 = r8367863 + r8367867;
        double r8367869 = r8367862 - r8367868;
        double r8367870 = r8367855 * r8367869;
        return r8367870;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.5
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied add-log-exp0.9

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

  6. Final simplification0.9

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

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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