Average Error: 43.2 → 0.9
Time: 9.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\left(\frac{1}{3} \cdot im\right) \cdot im\right) \cdot im + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\left(\frac{1}{3} \cdot im\right) \cdot im\right) \cdot im + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r164022 = 0.5;
        double r164023 = re;
        double r164024 = sin(r164023);
        double r164025 = r164022 * r164024;
        double r164026 = im;
        double r164027 = -r164026;
        double r164028 = exp(r164027);
        double r164029 = exp(r164026);
        double r164030 = r164028 - r164029;
        double r164031 = r164025 * r164030;
        return r164031;
}
↓
double f(double re, double im) {
        double r164032 = 0.5;
        double r164033 = re;
        double r164034 = sin(r164033);
        double r164035 = r164032 * r164034;
        double r164036 = 0.3333333333333333;
        double r164037 = im;
        double r164038 = r164036 * r164037;
        double r164039 = r164038 * r164037;
        double r164040 = r164039 * r164037;
        double r164041 = 0.016666666666666666;
        double r164042 = 5.0;
        double r164043 = pow(r164037, r164042);
        double r164044 = r164041 * r164043;
        double r164045 = 2.0;
        double r164046 = r164045 * r164037;
        double r164047 = r164044 + r164046;
        double r164048 = r164040 + r164047;
        double r164049 = -r164048;
        double r164050 = r164035 * r164049;
        return r164050;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

  4. Applied unpow30.9

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

  5. Applied associate-*r*0.9

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

  6. Simplified0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2020100 
(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))))