Average Error: 0.0 → 0.0
Time: 5.4s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r36827 = x;
        double r36828 = exp(r36827);
        double r36829 = -r36827;
        double r36830 = exp(r36829);
        double r36831 = r36828 + r36830;
        double r36832 = 2.0;
        double r36833 = r36831 / r36832;
        double r36834 = y;
        double r36835 = cos(r36834);
        double r36836 = r36833 * r36835;
        double r36837 = r36828 - r36830;
        double r36838 = r36837 / r36832;
        double r36839 = sin(r36834);
        double r36840 = r36838 * r36839;
        double r36841 = /* ERROR: no complex support in C */;
        double r36842 = /* ERROR: no complex support in C */;
        return r36842;
}
↓
double f(double x, double y) {
        double r36843 = x;
        double r36844 = exp(r36843);
        double r36845 = -r36843;
        double r36846 = exp(r36845);
        double r36847 = r36844 + r36846;
        double r36848 = 2.0;
        double r36849 = r36847 / r36848;
        double r36850 = y;
        double r36851 = cos(r36850);
        double r36852 = r36849 * r36851;
        return r36852;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} + e^{-x}}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))