Average Error: 58.0 → 0.8
Time: 10.5s
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 r133013 = 0.5;
        double r133014 = re;
        double r133015 = cos(r133014);
        double r133016 = r133013 * r133015;
        double r133017 = 0.0;
        double r133018 = im;
        double r133019 = r133017 - r133018;
        double r133020 = exp(r133019);
        double r133021 = exp(r133018);
        double r133022 = r133020 - r133021;
        double r133023 = r133016 * r133022;
        return r133023;
}
↓
double f(double re, double im) {
        double r133024 = 0.5;
        double r133025 = re;
        double r133026 = cos(r133025);
        double r133027 = r133024 * r133026;
        double r133028 = 0.3333333333333333;
        double r133029 = im;
        double r133030 = 3.0;
        double r133031 = pow(r133029, r133030);
        double r133032 = r133028 * r133031;
        double r133033 = 0.016666666666666666;
        double r133034 = 5.0;
        double r133035 = pow(r133029, r133034);
        double r133036 = r133033 * r133035;
        double r133037 = 2.0;
        double r133038 = r133037 * r133029;
        double r133039 = r133036 + r133038;
        double r133040 = r133032 + r133039;
        double r133041 = -r133040;
        double r133042 = r133027 * r133041;
        return r133042;
}

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
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2019353 
(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))))