Average Error: 58.1 → 0.7
Time: 10.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[-\left(0.166666666666666657 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.00833333333333333322 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
-\left(0.166666666666666657 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.00833333333333333322 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)
double f(double re, double im) {
        double r175907 = 0.5;
        double r175908 = re;
        double r175909 = cos(r175908);
        double r175910 = r175907 * r175909;
        double r175911 = 0.0;
        double r175912 = im;
        double r175913 = r175911 - r175912;
        double r175914 = exp(r175913);
        double r175915 = exp(r175912);
        double r175916 = r175914 - r175915;
        double r175917 = r175910 * r175916;
        return r175917;
}
↓
double f(double re, double im) {
        double r175918 = 0.16666666666666666;
        double r175919 = re;
        double r175920 = cos(r175919);
        double r175921 = im;
        double r175922 = 3.0;
        double r175923 = pow(r175921, r175922);
        double r175924 = r175920 * r175923;
        double r175925 = r175918 * r175924;
        double r175926 = 0.008333333333333333;
        double r175927 = 5.0;
        double r175928 = pow(r175921, r175927);
        double r175929 = r175920 * r175928;
        double r175930 = r175926 * r175929;
        double r175931 = 1.0;
        double r175932 = r175920 * r175921;
        double r175933 = r175931 * r175932;
        double r175934 = r175930 + r175933;
        double r175935 = r175925 + r175934;
        double r175936 = -r175935;
        return r175936;
}

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
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Taylor expanded around inf 0.7

    \[\leadsto \color{blue}{-\left(0.166666666666666657 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.00833333333333333322 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto -\left(0.166666666666666657 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(0.00833333333333333322 \cdot \left(\cos re \cdot {im}^{5}\right) + 1 \cdot \left(\cos re \cdot im\right)\right)\right)\]

Reproduce

herbie shell --seed 2020036 
(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))))