Average Error: 43.1 → 0.8
Time: 33.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r49877 = x;
        double r49878 = exp(r49877);
        double r49879 = -r49877;
        double r49880 = exp(r49879);
        double r49881 = r49878 + r49880;
        double r49882 = 2.0;
        double r49883 = r49881 / r49882;
        double r49884 = y;
        double r49885 = cos(r49884);
        double r49886 = r49883 * r49885;
        double r49887 = r49878 - r49880;
        double r49888 = r49887 / r49882;
        double r49889 = sin(r49884);
        double r49890 = r49888 * r49889;
        double r49891 = /* ERROR: no complex support in C */;
        double r49892 = /* ERROR: no complex support in C */;
        return r49892;
}
↓
double f(double x, double y) {
        double r49893 = 0.3333333333333333;
        double r49894 = x;
        double r49895 = 3.0;
        double r49896 = pow(r49894, r49895);
        double r49897 = r49893 * r49896;
        double r49898 = 0.016666666666666666;
        double r49899 = 5.0;
        double r49900 = pow(r49894, r49899);
        double r49901 = r49898 * r49900;
        double r49902 = 2.0;
        double r49903 = r49902 * r49894;
        double r49904 = r49901 + r49903;
        double r49905 = r49897 + r49904;
        double r49906 = 2.0;
        double r49907 = r49905 / r49906;
        double r49908 = y;
        double r49909 = sin(r49908);
        double r49910 = r49907 * r49909;
        return r49910;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Simplified43.1

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019303 
(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)))))