Average Error: 58.0 → 0.8
Time: 11.5s
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 r173910 = 0.5;
        double r173911 = re;
        double r173912 = cos(r173911);
        double r173913 = r173910 * r173912;
        double r173914 = 0.0;
        double r173915 = im;
        double r173916 = r173914 - r173915;
        double r173917 = exp(r173916);
        double r173918 = exp(r173915);
        double r173919 = r173917 - r173918;
        double r173920 = r173913 * r173919;
        return r173920;
}
↓
double f(double re, double im) {
        double r173921 = 0.5;
        double r173922 = re;
        double r173923 = cos(r173922);
        double r173924 = r173921 * r173923;
        double r173925 = 0.3333333333333333;
        double r173926 = im;
        double r173927 = 3.0;
        double r173928 = pow(r173926, r173927);
        double r173929 = r173925 * r173928;
        double r173930 = 0.016666666666666666;
        double r173931 = 5.0;
        double r173932 = pow(r173926, r173931);
        double r173933 = r173930 * r173932;
        double r173934 = 2.0;
        double r173935 = r173934 * r173926;
        double r173936 = r173933 + r173935;
        double r173937 = r173929 + r173936;
        double r173938 = -r173937;
        double r173939 = r173924 * r173938;
        return r173939;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
Target0.3
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 58.0

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

  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 2020047 
(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))))