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

↓

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

double f(double x, double y) {
        double r1345571 = x;
        double r1345572 = exp(r1345571);
        double r1345573 = -r1345571;
        double r1345574 = exp(r1345573);
        double r1345575 = r1345572 + r1345574;
        double r1345576 = 2.0;
        double r1345577 = r1345575 / r1345576;
        double r1345578 = y;
        double r1345579 = cos(r1345578);
        double r1345580 = r1345577 * r1345579;
        double r1345581 = r1345572 - r1345574;
        double r1345582 = r1345581 / r1345576;
        double r1345583 = sin(r1345578);
        double r1345584 = r1345582 * r1345583;
        double r1345585 = /* ERROR: no complex support in C */;
        double r1345586 = /* ERROR: no complex support in C */;
        return r1345586;
}

↓

double f(double x, double y) {
        double r1345587 = x;
        double r1345588 = exp(r1345587);
        double r1345589 = y;
        double r1345590 = cos(r1345589);
        double r1345591 = r1345588 * r1345590;
        double r1345592 = r1345590 / r1345588;
        double r1345593 = r1345591 + r1345592;
        double r1345594 = 2.0;
        double r1345595 = r1345593 / r1345594;
        return r1345595;
}

\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}

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 2019101 
(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