Average Error: 43.1 → 0.8
Time: 26.1s
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 r4924924 = 0.5;
        double r4924925 = re;
        double r4924926 = sin(r4924925);
        double r4924927 = r4924924 * r4924926;
        double r4924928 = im;
        double r4924929 = -r4924928;
        double r4924930 = exp(r4924929);
        double r4924931 = exp(r4924928);
        double r4924932 = r4924930 - r4924931;
        double r4924933 = r4924927 * r4924932;
        return r4924933;
}
↓
double f(double re, double im) {
        double r4924934 = im;
        double r4924935 = r4924934 * r4924934;
        double r4924936 = r4924934 * r4924935;
        double r4924937 = -0.3333333333333333;
        double r4924938 = r4924936 * r4924937;
        double r4924939 = 5.0;
        double r4924940 = pow(r4924934, r4924939);
        double r4924941 = 0.016666666666666666;
        double r4924942 = r4924940 * r4924941;
        double r4924943 = r4924934 + r4924934;
        double r4924944 = r4924942 + r4924943;
        double r4924945 = r4924938 - r4924944;
        double r4924946 = 0.5;
        double r4924947 = re;
        double r4924948 = sin(r4924947);
        double r4924949 = r4924946 * r4924948;
        double r4924950 = r4924945 * r4924949;
        return r4924950;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
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 43.1

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019156 
(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))))