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

↓

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

\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} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}

double f(double x, double y) {
        double r732836 = x;
        double r732837 = exp(r732836);
        double r732838 = -r732836;
        double r732839 = exp(r732838);
        double r732840 = r732837 + r732839;
        double r732841 = 2.0;
        double r732842 = r732840 / r732841;
        double r732843 = y;
        double r732844 = cos(r732843);
        double r732845 = r732842 * r732844;
        double r732846 = r732837 - r732839;
        double r732847 = r732846 / r732841;
        double r732848 = sin(r732843);
        double r732849 = r732847 * r732848;
        double r732850 = /* ERROR: no complex support in C */;
        double r732851 = /* ERROR: no complex support in C */;
        return r732851;
}

↓

double f(double x, double y) {
        double r732852 = x;
        double r732853 = exp(r732852);
        double r732854 = y;
        double r732855 = cos(r732854);
        double r732856 = r732853 * r732855;
        double r732857 = r732855 / r732853;
        double r732858 = r732856 + r732857;
        double r732859 = 2.0;
        double r732860 = r732858 / r732859;
        return r732860;
}

Derivation

Initial program 0.0
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
Simplified0.0
\[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]
Final simplification0.0
\[\leadsto \frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))

Error

Derivation

Reproduce