Average Error: 58.1 → 0.8
Time: 10.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos 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 r211866 = 0.5;
        double r211867 = re;
        double r211868 = cos(r211867);
        double r211869 = r211866 * r211868;
        double r211870 = 0.0;
        double r211871 = im;
        double r211872 = r211870 - r211871;
        double r211873 = exp(r211872);
        double r211874 = exp(r211871);
        double r211875 = r211873 - r211874;
        double r211876 = r211869 * r211875;
        return r211876;
}
↓
double f(double re, double im) {
        double r211877 = 0.5;
        double r211878 = re;
        double r211879 = cos(r211878);
        double r211880 = r211877 * r211879;
        double r211881 = 0.3333333333333333;
        double r211882 = im;
        double r211883 = 3.0;
        double r211884 = pow(r211882, r211883);
        double r211885 = r211881 * r211884;
        double r211886 = 0.016666666666666666;
        double r211887 = 5.0;
        double r211888 = pow(r211882, r211887);
        double r211889 = r211886 * r211888;
        double r211890 = 2.0;
        double r211891 = r211890 * r211882;
        double r211892 = r211889 + r211891;
        double r211893 = r211885 + r211892;
        double r211894 = -r211893;
        double r211895 = r211880 * r211894;
        return r211895;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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