Average Error: 58.0 → 0.7
Time: 10.4s
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 r214820 = 0.5;
        double r214821 = re;
        double r214822 = cos(r214821);
        double r214823 = r214820 * r214822;
        double r214824 = 0.0;
        double r214825 = im;
        double r214826 = r214824 - r214825;
        double r214827 = exp(r214826);
        double r214828 = exp(r214825);
        double r214829 = r214827 - r214828;
        double r214830 = r214823 * r214829;
        return r214830;
}
↓
double f(double re, double im) {
        double r214831 = 0.5;
        double r214832 = re;
        double r214833 = cos(r214832);
        double r214834 = r214831 * r214833;
        double r214835 = 0.3333333333333333;
        double r214836 = im;
        double r214837 = 3.0;
        double r214838 = pow(r214836, r214837);
        double r214839 = r214835 * r214838;
        double r214840 = 0.016666666666666666;
        double r214841 = 5.0;
        double r214842 = pow(r214836, r214841);
        double r214843 = r214840 * r214842;
        double r214844 = 2.0;
        double r214845 = r214844 * r214836;
        double r214846 = r214843 + r214845;
        double r214847 = r214839 + r214846;
        double r214848 = -r214847;
        double r214849 = r214834 * r214848;
        return r214849;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2020057 
(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))))