Average Error: 43.2 → 0.7
Time: 30.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r4480120 = 0.5;
        double r4480121 = re;
        double r4480122 = sin(r4480121);
        double r4480123 = r4480120 * r4480122;
        double r4480124 = im;
        double r4480125 = -r4480124;
        double r4480126 = exp(r4480125);
        double r4480127 = exp(r4480124);
        double r4480128 = r4480126 - r4480127;
        double r4480129 = r4480123 * r4480128;
        return r4480129;
}
↓
double f(double re, double im) {
        double r4480130 = im;
        double r4480131 = r4480130 * r4480130;
        double r4480132 = r4480130 * r4480131;
        double r4480133 = -0.3333333333333333;
        double r4480134 = r4480132 * r4480133;
        double r4480135 = 5.0;
        double r4480136 = pow(r4480130, r4480135);
        double r4480137 = 0.016666666666666666;
        double r4480138 = r4480136 * r4480137;
        double r4480139 = r4480130 + r4480130;
        double r4480140 = r4480138 + r4480139;
        double r4480141 = r4480134 - r4480140;
        double r4480142 = 0.5;
        double r4480143 = re;
        double r4480144 = sin(r4480143);
        double r4480145 = r4480142 * r4480144;
        double r4480146 = r4480141 * r4480145;
        return r4480146;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 
(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))))