Average Error: 0.0 → 0.0
Time: 11.0s
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 r153413 = x;
        double r153414 = exp(r153413);
        double r153415 = -r153413;
        double r153416 = exp(r153415);
        double r153417 = r153414 + r153416;
        double r153418 = 2.0;
        double r153419 = r153417 / r153418;
        double r153420 = y;
        double r153421 = cos(r153420);
        double r153422 = r153419 * r153421;
        double r153423 = r153414 - r153416;
        double r153424 = r153423 / r153418;
        double r153425 = sin(r153420);
        double r153426 = r153424 * r153425;
        double r153427 = /* ERROR: no complex support in C */;
        double r153428 = /* ERROR: no complex support in C */;
        return r153428;
}
↓
double f(double x, double y) {
        double r153429 = x;
        double r153430 = exp(r153429);
        double r153431 = -r153429;
        double r153432 = exp(r153431);
        double r153433 = r153430 + r153432;
        double r153434 = 2.0;
        double r153435 = r153433 / r153434;
        double r153436 = y;
        double r153437 = cos(r153436);
        double r153438 = r153435 * r153437;
        double r153439 = r153430 - r153432;
        double r153440 = r153439 / r153434;
        double r153441 = sin(r153436);
        double r153442 = r153440 * r153441;
        double r153443 = /* ERROR: no complex support in C */;
        double r153444 = /* ERROR: no complex support in C */;
        return r153444;
}

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 2019155 +o rules:numerics
(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)))))