Average Error: 58.0 → 0.9
Time: 1.1m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r5722037 = 0.5;
        double r5722038 = re;
        double r5722039 = cos(r5722038);
        double r5722040 = r5722037 * r5722039;
        double r5722041 = 0.0;
        double r5722042 = im;
        double r5722043 = r5722041 - r5722042;
        double r5722044 = exp(r5722043);
        double r5722045 = exp(r5722042);
        double r5722046 = r5722044 - r5722045;
        double r5722047 = r5722040 * r5722046;
        return r5722047;
}
↓
double f(double re, double im) {
        double r5722048 = -0.016666666666666666;
        double r5722049 = im;
        double r5722050 = 5.0;
        double r5722051 = pow(r5722049, r5722050);
        double r5722052 = r5722048 * r5722051;
        double r5722053 = r5722049 + r5722049;
        double r5722054 = r5722052 - r5722053;
        double r5722055 = r5722049 * r5722049;
        double r5722056 = 0.3333333333333333;
        double r5722057 = r5722049 * r5722056;
        double r5722058 = r5722055 * r5722057;
        double r5722059 = r5722054 - r5722058;
        double r5722060 = 0.5;
        double r5722061 = re;
        double r5722062 = cos(r5722061);
        double r5722063 = r5722060 * r5722062;
        double r5722064 = r5722059 * r5722063;
        return r5722064;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1.0:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019165 
(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))))