Average Error: 43.2 → 0.8
Time: 34.7s
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 r53842 = x;
        double r53843 = exp(r53842);
        double r53844 = -r53842;
        double r53845 = exp(r53844);
        double r53846 = r53843 + r53845;
        double r53847 = 2.0;
        double r53848 = r53846 / r53847;
        double r53849 = y;
        double r53850 = cos(r53849);
        double r53851 = r53848 * r53850;
        double r53852 = r53843 - r53845;
        double r53853 = r53852 / r53847;
        double r53854 = sin(r53849);
        double r53855 = r53853 * r53854;
        double r53856 = /* ERROR: no complex support in C */;
        double r53857 = /* ERROR: no complex support in C */;
        return r53857;
}
↓
double f(double x, double y) {
        double r53858 = 0.3333333333333333;
        double r53859 = x;
        double r53860 = 3.0;
        double r53861 = pow(r53859, r53860);
        double r53862 = r53858 * r53861;
        double r53863 = 0.016666666666666666;
        double r53864 = 5.0;
        double r53865 = pow(r53859, r53864);
        double r53866 = r53863 * r53865;
        double r53867 = 2.0;
        double r53868 = r53867 * r53859;
        double r53869 = r53866 + r53868;
        double r53870 = r53862 + r53869;
        double r53871 = 2.0;
        double r53872 = r53870 / r53871;
        double r53873 = y;
        double r53874 = sin(r53873);
        double r53875 = r53872 * r53874;
        return r53875;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

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

  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 2019325 
(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)))))