Average Error: 43.8 → 0.8
Time: 8.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[-\left(0.1666666666666666574148081281236954964697 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333217685101601546193705872 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
-\left(0.1666666666666666574148081281236954964697 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333217685101601546193705872 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)
double f(double re, double im) {
        double r240127 = 0.5;
        double r240128 = re;
        double r240129 = sin(r240128);
        double r240130 = r240127 * r240129;
        double r240131 = im;
        double r240132 = -r240131;
        double r240133 = exp(r240132);
        double r240134 = exp(r240131);
        double r240135 = r240133 - r240134;
        double r240136 = r240130 * r240135;
        return r240136;
}
↓
double f(double re, double im) {
        double r240137 = 0.16666666666666666;
        double r240138 = re;
        double r240139 = sin(r240138);
        double r240140 = im;
        double r240141 = 3.0;
        double r240142 = pow(r240140, r240141);
        double r240143 = r240139 * r240142;
        double r240144 = r240137 * r240143;
        double r240145 = 1.0;
        double r240146 = r240139 * r240140;
        double r240147 = r240145 * r240146;
        double r240148 = 0.008333333333333333;
        double r240149 = 5.0;
        double r240150 = pow(r240140, r240149);
        double r240151 = r240139 * r240150;
        double r240152 = r240148 * r240151;
        double r240153 = r240147 + r240152;
        double r240154 = r240144 + r240153;
        double r240155 = -r240154;
        return r240155;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.8
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.8

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

  2. Taylor expanded around 0 0.8

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

  3. Taylor expanded around inf 0.8

    \[\leadsto \color{blue}{-\left(0.1666666666666666574148081281236954964697 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333217685101601546193705872 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  4. Final simplification0.8

    \[\leadsto -\left(0.1666666666666666574148081281236954964697 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333217685101601546193705872 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))