Average Error: 0.0 → 0.0
Time: 12.0s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r31374 = x;
        double r31375 = exp(r31374);
        double r31376 = -r31374;
        double r31377 = exp(r31376);
        double r31378 = r31375 + r31377;
        double r31379 = 2.0;
        double r31380 = r31378 / r31379;
        double r31381 = y;
        double r31382 = cos(r31381);
        double r31383 = r31380 * r31382;
        double r31384 = r31375 - r31377;
        double r31385 = r31384 / r31379;
        double r31386 = sin(r31381);
        double r31387 = r31385 * r31386;
        double r31388 = /* ERROR: no complex support in C */;
        double r31389 = /* ERROR: no complex support in C */;
        return r31389;
}
↓
double f(double x, double y) {
        double r31390 = x;
        double r31391 = exp(r31390);
        double r31392 = -r31390;
        double r31393 = exp(r31392);
        double r31394 = r31391 + r31393;
        double r31395 = 2.0;
        double r31396 = r31394 / r31395;
        double r31397 = y;
        double r31398 = cos(r31397);
        double r31399 = r31396 * r31398;
        double r31400 = r31391 - r31393;
        double r31401 = r31400 / r31395;
        double r31402 = sin(r31397);
        double r31403 = r31401 * r31402;
        double r31404 = /* ERROR: no complex support in C */;
        double r31405 = /* ERROR: no complex support in C */;
        return r31405;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 
(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)))))