Average Error: 43.6 → 0.7
Time: 30.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(-\left(\left(im \cdot \sin re\right) \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right) - \left(\left({im}^{5} \cdot \sin re\right) \cdot 0.008333333333333333 + \left(\sin re \cdot 1.0\right) \cdot im\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(-\left(\left(im \cdot \sin re\right) \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right) - \left(\left({im}^{5} \cdot \sin re\right) \cdot 0.008333333333333333 + \left(\sin re \cdot 1.0\right) \cdot im\right)
double f(double re, double im) {
        double r6269909 = 0.5;
        double r6269910 = re;
        double r6269911 = sin(r6269910);
        double r6269912 = r6269909 * r6269911;
        double r6269913 = im;
        double r6269914 = -r6269913;
        double r6269915 = exp(r6269914);
        double r6269916 = exp(r6269913);
        double r6269917 = r6269915 - r6269916;
        double r6269918 = r6269912 * r6269917;
        return r6269918;
}
↓
double f(double re, double im) {
        double r6269919 = im;
        double r6269920 = re;
        double r6269921 = sin(r6269920);
        double r6269922 = r6269919 * r6269921;
        double r6269923 = r6269919 * r6269919;
        double r6269924 = r6269922 * r6269923;
        double r6269925 = 0.16666666666666666;
        double r6269926 = r6269924 * r6269925;
        double r6269927 = -r6269926;
        double r6269928 = 5.0;
        double r6269929 = pow(r6269919, r6269928);
        double r6269930 = r6269929 * r6269921;
        double r6269931 = 0.008333333333333333;
        double r6269932 = r6269930 * r6269931;
        double r6269933 = 1.0;
        double r6269934 = r6269921 * r6269933;
        double r6269935 = r6269934 * r6269919;
        double r6269936 = r6269932 + r6269935;
        double r6269937 = r6269927 - r6269936;
        return r6269937;
}

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

Derivation

  1. Initial program 43.6

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

  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

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

  4. Taylor expanded around -inf 0.7

    \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  5. Simplified0.7

    \[\leadsto \color{blue}{\left(\left(\sin re \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(-0.16666666666666666\right) - \left(\left({im}^{5} \cdot \sin re\right) \cdot 0.008333333333333333 + im \cdot \left(1.0 \cdot \sin re\right)\right)}\]

  6. Final simplification0.7

    \[\leadsto \left(-\left(\left(im \cdot \sin re\right) \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right) - \left(\left({im}^{5} \cdot \sin re\right) \cdot 0.008333333333333333 + \left(\sin re \cdot 1.0\right) \cdot im\right)\]

Reproduce

herbie shell --seed 2019132 
(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))))