Average Error: 43.4 → 0.9
Time: 17.0s
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 r80830 = x;
        double r80831 = exp(r80830);
        double r80832 = -r80830;
        double r80833 = exp(r80832);
        double r80834 = r80831 + r80833;
        double r80835 = 2.0;
        double r80836 = r80834 / r80835;
        double r80837 = y;
        double r80838 = cos(r80837);
        double r80839 = r80836 * r80838;
        double r80840 = r80831 - r80833;
        double r80841 = r80840 / r80835;
        double r80842 = sin(r80837);
        double r80843 = r80841 * r80842;
        double r80844 = /* ERROR: no complex support in C */;
        double r80845 = /* ERROR: no complex support in C */;
        return r80845;
}
↓
double f(double x, double y) {
        double r80846 = 0.3333333333333333;
        double r80847 = x;
        double r80848 = 3.0;
        double r80849 = pow(r80847, r80848);
        double r80850 = r80846 * r80849;
        double r80851 = 0.016666666666666666;
        double r80852 = 5.0;
        double r80853 = pow(r80847, r80852);
        double r80854 = r80851 * r80853;
        double r80855 = 2.0;
        double r80856 = r80855 * r80847;
        double r80857 = r80854 + r80856;
        double r80858 = r80850 + r80857;
        double r80859 = 2.0;
        double r80860 = r80858 / r80859;
        double r80861 = y;
        double r80862 = sin(r80861);
        double r80863 = r80860 * r80862;
        return r80863;
}

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. Simplified43.4

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

  3. Taylor expanded around 0 0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019318 
(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)))))