Average Error: 58.4 → 0.7
Time: 37.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]
\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(\frac{-1}{3} \cdot {im}^{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r186092 = 0.5;
        double r186093 = re;
        double r186094 = cos(r186093);
        double r186095 = r186092 * r186094;
        double r186096 = 0.0;
        double r186097 = im;
        double r186098 = r186096 - r186097;
        double r186099 = exp(r186098);
        double r186100 = exp(r186097);
        double r186101 = r186099 - r186100;
        double r186102 = r186095 * r186101;
        return r186102;
}
↓
double f(double re, double im) {
        double r186103 = 0.5;
        double r186104 = re;
        double r186105 = cos(r186104);
        double r186106 = r186103 * r186105;
        double r186107 = -0.3333333333333333;
        double r186108 = im;
        double r186109 = 3.0;
        double r186110 = pow(r186108, r186109);
        double r186111 = r186107 * r186110;
        double r186112 = 0.016666666666666666;
        double r186113 = 5.0;
        double r186114 = pow(r186108, r186113);
        double r186115 = r186112 * r186114;
        double r186116 = 2.0;
        double r186117 = r186116 * r186108;
        double r186118 = r186115 + r186117;
        double r186119 = r186111 - r186118;
        double r186120 = r186106 * r186119;
        return r186120;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
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 58.4

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

  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. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019199 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))