Average Error: 43.6 → 0.7
Time: 18.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r163836 = 0.5;
        double r163837 = re;
        double r163838 = sin(r163837);
        double r163839 = r163836 * r163838;
        double r163840 = im;
        double r163841 = -r163840;
        double r163842 = exp(r163841);
        double r163843 = exp(r163840);
        double r163844 = r163842 - r163843;
        double r163845 = r163839 * r163844;
        return r163845;
}
↓
double f(double re, double im) {
        double r163846 = 0.5;
        double r163847 = re;
        double r163848 = sin(r163847);
        double r163849 = r163846 * r163848;
        double r163850 = 0.3333333333333333;
        double r163851 = im;
        double r163852 = 3.0;
        double r163853 = pow(r163851, r163852);
        double r163854 = r163850 * r163853;
        double r163855 = 0.016666666666666666;
        double r163856 = 5.0;
        double r163857 = pow(r163851, r163856);
        double r163858 = r163855 * r163857;
        double r163859 = r163854 + r163858;
        double r163860 = 2.0;
        double r163861 = r163860 * r163851;
        double r163862 = r163859 + r163861;
        double r163863 = -r163862;
        double r163864 = r163849 * r163863;
        return r163864;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.6
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.6

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

  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. Using strategy rm

  4. Applied associate-+r+0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019308 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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