Average Error: 43.6 → 0.8
Time: 27.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot {im}^{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot {im}^{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 r262021 = 0.5;
        double r262022 = re;
        double r262023 = sin(r262022);
        double r262024 = r262021 * r262023;
        double r262025 = im;
        double r262026 = -r262025;
        double r262027 = exp(r262026);
        double r262028 = exp(r262025);
        double r262029 = r262027 - r262028;
        double r262030 = r262024 * r262029;
        return r262030;
}
↓
double f(double re, double im) {
        double r262031 = -0.3333333333333333;
        double r262032 = im;
        double r262033 = 3.0;
        double r262034 = pow(r262032, r262033);
        double r262035 = r262031 * r262034;
        double r262036 = 5.0;
        double r262037 = pow(r262032, r262036);
        double r262038 = 0.016666666666666666;
        double r262039 = r262037 * r262038;
        double r262040 = r262032 + r262032;
        double r262041 = r262039 + r262040;
        double r262042 = r262035 - r262041;
        double r262043 = 0.5;
        double r262044 = re;
        double r262045 = sin(r262044);
        double r262046 = r262043 * r262045;
        double r262047 = r262042 * r262046;
        return r262047;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.6
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 43.6

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))