Average Error: 57.9 → 0.8
Time: 42.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7929941 = 0.5;
        double r7929942 = re;
        double r7929943 = cos(r7929942);
        double r7929944 = r7929941 * r7929943;
        double r7929945 = 0.0;
        double r7929946 = im;
        double r7929947 = r7929945 - r7929946;
        double r7929948 = exp(r7929947);
        double r7929949 = exp(r7929946);
        double r7929950 = r7929948 - r7929949;
        double r7929951 = r7929944 * r7929950;
        return r7929951;
}
↓
double f(double re, double im) {
        double r7929952 = -0.3333333333333333;
        double r7929953 = im;
        double r7929954 = r7929953 * r7929953;
        double r7929955 = r7929953 * r7929954;
        double r7929956 = r7929952 * r7929955;
        double r7929957 = 5.0;
        double r7929958 = pow(r7929953, r7929957);
        double r7929959 = -0.016666666666666666;
        double r7929960 = r7929958 * r7929959;
        double r7929961 = r7929953 + r7929953;
        double r7929962 = r7929960 - r7929961;
        double r7929963 = r7929956 + r7929962;
        double r7929964 = 0.5;
        double r7929965 = re;
        double r7929966 = cos(r7929965);
        double r7929967 = r7929964 * r7929966;
        double r7929968 = r7929963 * r7929967;
        return r7929968;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))