Average Error: 43.8 → 0.7
Time: 11.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r40838 = x;
        double r40839 = exp(r40838);
        double r40840 = -r40838;
        double r40841 = exp(r40840);
        double r40842 = r40839 + r40841;
        double r40843 = 2.0;
        double r40844 = r40842 / r40843;
        double r40845 = y;
        double r40846 = cos(r40845);
        double r40847 = r40844 * r40846;
        double r40848 = r40839 - r40841;
        double r40849 = r40848 / r40843;
        double r40850 = sin(r40845);
        double r40851 = r40849 * r40850;
        double r40852 = /* ERROR: no complex support in C */;
        double r40853 = /* ERROR: no complex support in C */;
        return r40853;
}
↓
double f(double x, double y) {
        double r40854 = x;
        double r40855 = exp(r40854);
        double r40856 = -r40854;
        double r40857 = exp(r40856);
        double r40858 = r40855 + r40857;
        double r40859 = 2.0;
        double r40860 = r40858 / r40859;
        double r40861 = y;
        double r40862 = cos(r40861);
        double r40863 = r40860 * r40862;
        double r40864 = 0.3333333333333333;
        double r40865 = 3.0;
        double r40866 = pow(r40854, r40865);
        double r40867 = r40864 * r40866;
        double r40868 = 0.016666666666666666;
        double r40869 = 5.0;
        double r40870 = pow(r40854, r40869);
        double r40871 = r40868 * r40870;
        double r40872 = r40867 + r40871;
        double r40873 = 2.0;
        double r40874 = r40873 * r40854;
        double r40875 = r40872 + r40874;
        double r40876 = r40875 / r40859;
        double r40877 = sin(r40861);
        double r40878 = r40876 * r40877;
        double r40879 = /* ERROR: no complex support in C */;
        double r40880 = /* ERROR: no complex support in C */;
        return r40880;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y i\right))\]
  3. Using strategy rm

  4. Applied associate-+r+0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}}{2} \cdot \sin y i\right))\]

  5. Final simplification0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))