Average Error: 0.0 → 0.0
Time: 4.8s
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 r58439 = x;
        double r58440 = exp(r58439);
        double r58441 = -r58439;
        double r58442 = exp(r58441);
        double r58443 = r58440 + r58442;
        double r58444 = 2.0;
        double r58445 = r58443 / r58444;
        double r58446 = y;
        double r58447 = cos(r58446);
        double r58448 = r58445 * r58447;
        double r58449 = r58440 - r58442;
        double r58450 = r58449 / r58444;
        double r58451 = sin(r58446);
        double r58452 = r58450 * r58451;
        double r58453 = /* ERROR: no complex support in C */;
        double r58454 = /* ERROR: no complex support in C */;
        return r58454;
}
↓
double f(double x, double y) {
        double r58455 = x;
        double r58456 = exp(r58455);
        double r58457 = -r58455;
        double r58458 = exp(r58457);
        double r58459 = r58456 + r58458;
        double r58460 = 2.0;
        double r58461 = r58459 / r58460;
        double r58462 = y;
        double r58463 = cos(r58462);
        double r58464 = r58461 * r58463;
        double r58465 = r58456 - r58458;
        double r58466 = r58465 / r58460;
        double r58467 = sin(r58462);
        double r58468 = r58466 * r58467;
        double r58469 = /* ERROR: no complex support in C */;
        double r58470 = /* ERROR: no complex support in C */;
        return r58470;
}

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 2020064 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))