Average Error: 43.5 → 0.9
Time: 35.6s
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 r10091013 = 0.5;
        double r10091014 = re;
        double r10091015 = sin(r10091014);
        double r10091016 = r10091013 * r10091015;
        double r10091017 = im;
        double r10091018 = -r10091017;
        double r10091019 = exp(r10091018);
        double r10091020 = exp(r10091017);
        double r10091021 = r10091019 - r10091020;
        double r10091022 = r10091016 * r10091021;
        return r10091022;
}
↓
double f(double re, double im) {
        double r10091023 = 0.5;
        double r10091024 = re;
        double r10091025 = sin(r10091024);
        double r10091026 = r10091023 * r10091025;
        double r10091027 = im;
        double r10091028 = r10091027 * r10091027;
        double r10091029 = r10091027 * r10091028;
        double r10091030 = -0.3333333333333333;
        double r10091031 = r10091029 * r10091030;
        double r10091032 = 2.0;
        double r10091033 = r10091032 * r10091027;
        double r10091034 = 0.016666666666666666;
        double r10091035 = 5.0;
        double r10091036 = pow(r10091027, r10091035);
        double r10091037 = r10091034 * r10091036;
        double r10091038 = r10091033 + r10091037;
        double r10091039 = r10091031 - r10091038;
        double r10091040 = r10091026 * r10091039;
        return r10091040;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.5
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019137 
(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))))