Average Error: 43.2 → 0.8
Time: 38.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(2 \cdot im + \frac{1}{60} \cdot {im}^{5}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(2 \cdot im + \frac{1}{60} \cdot {im}^{5}\right)\right)
double f(double re, double im) {
        double r9033127 = 0.5;
        double r9033128 = re;
        double r9033129 = sin(r9033128);
        double r9033130 = r9033127 * r9033129;
        double r9033131 = im;
        double r9033132 = -r9033131;
        double r9033133 = exp(r9033132);
        double r9033134 = exp(r9033131);
        double r9033135 = r9033133 - r9033134;
        double r9033136 = r9033130 * r9033135;
        return r9033136;
}
↓
double f(double re, double im) {
        double r9033137 = 0.5;
        double r9033138 = re;
        double r9033139 = sin(r9033138);
        double r9033140 = r9033137 * r9033139;
        double r9033141 = im;
        double r9033142 = r9033141 * r9033141;
        double r9033143 = r9033141 * r9033142;
        double r9033144 = -0.3333333333333333;
        double r9033145 = r9033143 * r9033144;
        double r9033146 = 2.0;
        double r9033147 = r9033146 * r9033141;
        double r9033148 = 0.016666666666666666;
        double r9033149 = 5.0;
        double r9033150 = pow(r9033141, r9033149);
        double r9033151 = r9033148 * r9033150;
        double r9033152 = r9033147 + r9033151;
        double r9033153 = r9033145 - r9033152;
        double r9033154 = r9033140 * r9033153;
        return r9033154;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.2
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.2

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

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019135 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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