Average Error: 58.0 → 0.7
Time: 25.1s
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 r264160 = 0.5;
        double r264161 = re;
        double r264162 = cos(r264161);
        double r264163 = r264160 * r264162;
        double r264164 = 0.0;
        double r264165 = im;
        double r264166 = r264164 - r264165;
        double r264167 = exp(r264166);
        double r264168 = exp(r264165);
        double r264169 = r264167 - r264168;
        double r264170 = r264163 * r264169;
        return r264170;
}
↓
double f(double re, double im) {
        double r264171 = 0.5;
        double r264172 = re;
        double r264173 = cos(r264172);
        double r264174 = r264171 * r264173;
        double r264175 = -0.3333333333333333;
        double r264176 = im;
        double r264177 = 3.0;
        double r264178 = pow(r264176, r264177);
        double r264179 = r264175 * r264178;
        double r264180 = 0.016666666666666666;
        double r264181 = 5.0;
        double r264182 = pow(r264176, r264181);
        double r264183 = r264180 * r264182;
        double r264184 = 2.0;
        double r264185 = r264184 * r264176;
        double r264186 = r264183 + r264185;
        double r264187 = r264179 - r264186;
        double r264188 = r264174 * r264187;
        return r264188;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
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 58.0

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

  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 2020043 
(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))))