Average Error: 43.2 → 0.7
Time: 28.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot {im}^{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \sin re \cdot \left(\left(im \cdot -2 + \frac{-1}{60} \cdot {im}^{5}\right) \cdot 0.5\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot {im}^{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \sin re \cdot \left(\left(im \cdot -2 + \frac{-1}{60} \cdot {im}^{5}\right) \cdot 0.5\right)
double f(double re, double im) {
        double r283044 = 0.5;
        double r283045 = re;
        double r283046 = sin(r283045);
        double r283047 = r283044 * r283046;
        double r283048 = im;
        double r283049 = -r283048;
        double r283050 = exp(r283049);
        double r283051 = exp(r283048);
        double r283052 = r283050 - r283051;
        double r283053 = r283047 * r283052;
        return r283053;
}
↓
double f(double re, double im) {
        double r283054 = -0.3333333333333333;
        double r283055 = im;
        double r283056 = 3.0;
        double r283057 = pow(r283055, r283056);
        double r283058 = r283054 * r283057;
        double r283059 = 0.5;
        double r283060 = re;
        double r283061 = sin(r283060);
        double r283062 = r283059 * r283061;
        double r283063 = r283058 * r283062;
        double r283064 = -2.0;
        double r283065 = r283055 * r283064;
        double r283066 = -0.016666666666666666;
        double r283067 = 5.0;
        double r283068 = pow(r283055, r283067);
        double r283069 = r283066 * r283068;
        double r283070 = r283065 + r283069;
        double r283071 = r283070 * r283059;
        double r283072 = r283061 * r283071;
        double r283073 = r283063 + r283072;
        return r283073;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.2
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.2

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

  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

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Simplified0.7

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))