Average Error: 58.2 → 0.7
Time: 10.3s
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 r273130 = 0.5;
        double r273131 = re;
        double r273132 = cos(r273131);
        double r273133 = r273130 * r273132;
        double r273134 = 0.0;
        double r273135 = im;
        double r273136 = r273134 - r273135;
        double r273137 = exp(r273136);
        double r273138 = exp(r273135);
        double r273139 = r273137 - r273138;
        double r273140 = r273133 * r273139;
        return r273140;
}
↓
double f(double re, double im) {
        double r273141 = 0.5;
        double r273142 = re;
        double r273143 = cos(r273142);
        double r273144 = r273141 * r273143;
        double r273145 = 0.3333333333333333;
        double r273146 = im;
        double r273147 = 3.0;
        double r273148 = pow(r273146, r273147);
        double r273149 = r273145 * r273148;
        double r273150 = 0.016666666666666666;
        double r273151 = 5.0;
        double r273152 = pow(r273146, r273151);
        double r273153 = r273150 * r273152;
        double r273154 = 2.0;
        double r273155 = r273154 * r273146;
        double r273156 = r273153 + r273155;
        double r273157 = r273149 + r273156;
        double r273158 = -r273157;
        double r273159 = r273144 * r273158;
        return r273159;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
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 58.2

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

  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 2020059 
(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))))