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

↓

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

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

↓

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

double f(double x, double y) {
        double r761562 = x;
        double r761563 = exp(r761562);
        double r761564 = -r761562;
        double r761565 = exp(r761564);
        double r761566 = r761563 + r761565;
        double r761567 = 2.0;
        double r761568 = r761566 / r761567;
        double r761569 = y;
        double r761570 = cos(r761569);
        double r761571 = r761568 * r761570;
        double r761572 = r761563 - r761565;
        double r761573 = r761572 / r761567;
        double r761574 = sin(r761569);
        double r761575 = r761573 * r761574;
        double r761576 = /* ERROR: no complex support in C */;
        double r761577 = /* ERROR: no complex support in C */;
        return r761577;
}

↓

double f(double x, double y) {
        double r761578 = y;
        double r761579 = cos(r761578);
        double r761580 = x;
        double r761581 = exp(r761580);
        double r761582 = r761579 / r761581;
        double r761583 = r761581 * r761579;
        double r761584 = r761582 + r761583;
        double r761585 = 2.0;
        double r761586 = r761584 / r761585;
        return r761586;
}

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
\[\leadsto \color{blue}{\frac{\cos y \cdot e^{x} + \frac{\cos y}{e^{x}}}{2}}\]
Final simplification0.0
\[\leadsto \frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 2019134 
(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