Average Error: 58.2 → 0.7
Time: 10.9s
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 r172927 = 0.5;
        double r172928 = re;
        double r172929 = cos(r172928);
        double r172930 = r172927 * r172929;
        double r172931 = 0.0;
        double r172932 = im;
        double r172933 = r172931 - r172932;
        double r172934 = exp(r172933);
        double r172935 = exp(r172932);
        double r172936 = r172934 - r172935;
        double r172937 = r172930 * r172936;
        return r172937;
}
↓
double f(double re, double im) {
        double r172938 = 0.5;
        double r172939 = re;
        double r172940 = cos(r172939);
        double r172941 = r172938 * r172940;
        double r172942 = 0.3333333333333333;
        double r172943 = im;
        double r172944 = 3.0;
        double r172945 = pow(r172943, r172944);
        double r172946 = r172942 * r172945;
        double r172947 = 0.016666666666666666;
        double r172948 = 5.0;
        double r172949 = pow(r172943, r172948);
        double r172950 = r172947 * r172949;
        double r172951 = 2.0;
        double r172952 = r172951 * r172943;
        double r172953 = r172950 + r172952;
        double r172954 = r172946 + r172953;
        double r172955 = -r172954;
        double r172956 = r172941 * r172955;
        return r172956;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
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 58.2

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

  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2020003 
(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))))