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

↓

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

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

↓

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

double f(double x, double y) {
        double r19464 = x;
        double r19465 = exp(r19464);
        double r19466 = -r19464;
        double r19467 = exp(r19466);
        double r19468 = r19465 + r19467;
        double r19469 = 2.0;
        double r19470 = r19468 / r19469;
        double r19471 = y;
        double r19472 = cos(r19471);
        double r19473 = r19470 * r19472;
        double r19474 = r19465 - r19467;
        double r19475 = r19474 / r19469;
        double r19476 = sin(r19471);
        double r19477 = r19475 * r19476;
        double r19478 = /* ERROR: no complex support in C */;
        double r19479 = /* ERROR: no complex support in C */;
        return r19479;
}

↓

double f(double x, double y) {
        double r19480 = x;
        double r19481 = exp(r19480);
        double r19482 = -r19480;
        double r19483 = exp(r19482);
        double r19484 = r19481 + r19483;
        double r19485 = y;
        double r19486 = cos(r19485);
        double r19487 = 2.0;
        double r19488 = r19486 / r19487;
        double r19489 = r19484 * r19488;
        return r19489;
}

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))

Error

Derivation

Reproduce