Average Error: 43.4 → 0.9
Time: 2.4m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))\]
\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{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r7141839 = x;
        double r7141840 = exp(r7141839);
        double r7141841 = -r7141839;
        double r7141842 = exp(r7141841);
        double r7141843 = r7141840 + r7141842;
        double r7141844 = 2.0;
        double r7141845 = r7141843 / r7141844;
        double r7141846 = y;
        double r7141847 = cos(r7141846);
        double r7141848 = r7141845 * r7141847;
        double r7141849 = r7141840 - r7141842;
        double r7141850 = r7141849 / r7141844;
        double r7141851 = sin(r7141846);
        double r7141852 = r7141850 * r7141851;
        double r7141853 = /* ERROR: no complex support in C */;
        double r7141854 = /* ERROR: no complex support in C */;
        return r7141854;
}
↓
double f(double x, double y) {
        double r7141855 = x;
        double r7141856 = exp(r7141855);
        double r7141857 = -r7141855;
        double r7141858 = exp(r7141857);
        double r7141859 = r7141856 + r7141858;
        double r7141860 = 2.0;
        double r7141861 = r7141859 / r7141860;
        double r7141862 = y;
        double r7141863 = cos(r7141862);
        double r7141864 = r7141861 * r7141863;
        double r7141865 = 5.0;
        double r7141866 = pow(r7141855, r7141865);
        double r7141867 = 0.016666666666666666;
        double r7141868 = r7141866 * r7141867;
        double r7141869 = 0.3333333333333333;
        double r7141870 = r7141855 * r7141869;
        double r7141871 = r7141870 * r7141855;
        double r7141872 = r7141860 + r7141871;
        double r7141873 = r7141872 * r7141855;
        double r7141874 = r7141868 + r7141873;
        double r7141875 = r7141874 / r7141860;
        double r7141876 = sin(r7141862);
        double r7141877 = r7141875 * r7141876;
        double r7141878 = /* ERROR: no complex support in C */;
        double r7141879 = /* ERROR: no complex support in C */;
        return r7141879;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))