Average Error: 43.4 → 0.8
Time: 41.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right)}{2} i\right))\]
\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 + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right)}{2} i\right))
double f(double x, double y) {
        double r2101873 = x;
        double r2101874 = exp(r2101873);
        double r2101875 = -r2101873;
        double r2101876 = exp(r2101875);
        double r2101877 = r2101874 + r2101876;
        double r2101878 = 2.0;
        double r2101879 = r2101877 / r2101878;
        double r2101880 = y;
        double r2101881 = cos(r2101880);
        double r2101882 = r2101879 * r2101881;
        double r2101883 = r2101874 - r2101876;
        double r2101884 = r2101883 / r2101878;
        double r2101885 = sin(r2101880);
        double r2101886 = r2101884 * r2101885;
        double r2101887 = /* ERROR: no complex support in C */;
        double r2101888 = /* ERROR: no complex support in C */;
        return r2101888;
}
↓
double f(double x, double y) {
        double r2101889 = x;
        double r2101890 = exp(r2101889);
        double r2101891 = -r2101889;
        double r2101892 = exp(r2101891);
        double r2101893 = r2101890 + r2101892;
        double r2101894 = 2.0;
        double r2101895 = r2101893 / r2101894;
        double r2101896 = y;
        double r2101897 = cos(r2101896);
        double r2101898 = r2101895 * r2101897;
        double r2101899 = sin(r2101896);
        double r2101900 = 0.016666666666666666;
        double r2101901 = 5.0;
        double r2101902 = pow(r2101889, r2101901);
        double r2101903 = r2101900 * r2101902;
        double r2101904 = 2.0;
        double r2101905 = r2101889 * r2101904;
        double r2101906 = r2101889 * r2101889;
        double r2101907 = r2101906 * r2101889;
        double r2101908 = 0.3333333333333333;
        double r2101909 = r2101907 * r2101908;
        double r2101910 = r2101905 + r2101909;
        double r2101911 = r2101903 + r2101910;
        double r2101912 = r2101911 / r2101894;
        double r2101913 = r2101899 * r2101912;
        double r2101914 = /* ERROR: no complex support in C */;
        double r2101915 = /* ERROR: no complex support in C */;
        return r2101915;
}

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 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))