Average Error: 57.8 → 0.9
Time: 41.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) + \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) + \frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r8648157 = 0.5;
        double r8648158 = re;
        double r8648159 = cos(r8648158);
        double r8648160 = r8648157 * r8648159;
        double r8648161 = 0.0;
        double r8648162 = im;
        double r8648163 = r8648161 - r8648162;
        double r8648164 = exp(r8648163);
        double r8648165 = exp(r8648162);
        double r8648166 = r8648164 - r8648165;
        double r8648167 = r8648160 * r8648166;
        return r8648167;
}
↓
double f(double re, double im) {
        double r8648168 = -0.016666666666666666;
        double r8648169 = im;
        double r8648170 = 5.0;
        double r8648171 = pow(r8648169, r8648170);
        double r8648172 = r8648168 * r8648171;
        double r8648173 = r8648169 + r8648169;
        double r8648174 = r8648172 - r8648173;
        double r8648175 = -0.3333333333333333;
        double r8648176 = r8648169 * r8648169;
        double r8648177 = r8648169 * r8648176;
        double r8648178 = r8648175 * r8648177;
        double r8648179 = r8648174 + r8648178;
        double r8648180 = 0.5;
        double r8648181 = re;
        double r8648182 = cos(r8648181);
        double r8648183 = r8648180 * r8648182;
        double r8648184 = r8648179 * r8648183;
        return r8648184;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 57.8

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019143 
(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))))