Average Error: 43.4 → 0.9
Time: 1.1m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(\sin re \cdot 0.5\right) + \left(im \cdot \left(-2 - im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(\sin re \cdot 0.5\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(\sin re \cdot 0.5\right) + \left(im \cdot \left(-2 - im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(\sin re \cdot 0.5\right)
double f(double re, double im) {
        double r44848924 = 0.5;
        double r44848925 = re;
        double r44848926 = sin(r44848925);
        double r44848927 = r44848924 * r44848926;
        double r44848928 = im;
        double r44848929 = -r44848928;
        double r44848930 = exp(r44848929);
        double r44848931 = exp(r44848928);
        double r44848932 = r44848930 - r44848931;
        double r44848933 = r44848927 * r44848932;
        return r44848933;
}
↓
double f(double re, double im) {
        double r44848934 = im;
        double r44848935 = 5.0;
        double r44848936 = pow(r44848934, r44848935);
        double r44848937 = -0.016666666666666666;
        double r44848938 = r44848936 * r44848937;
        double r44848939 = re;
        double r44848940 = sin(r44848939);
        double r44848941 = 0.5;
        double r44848942 = r44848940 * r44848941;
        double r44848943 = r44848938 * r44848942;
        double r44848944 = -2.0;
        double r44848945 = 0.3333333333333333;
        double r44848946 = r44848945 * r44848934;
        double r44848947 = r44848934 * r44848946;
        double r44848948 = r44848944 - r44848947;
        double r44848949 = r44848934 * r44848948;
        double r44848950 = r44848949 * r44848942;
        double r44848951 = r44848943 + r44848950;
        return r44848951;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
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 43.4

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

  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

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

  5. Applied sub-neg0.9

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

  6. Applied distribute-rgt-in0.9

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

  7. Simplified0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2019124 
(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))))