Average Error: 0.0 → 0.0
Time: 22.0s
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 r36681 = x;
        double r36682 = exp(r36681);
        double r36683 = -r36681;
        double r36684 = exp(r36683);
        double r36685 = r36682 + r36684;
        double r36686 = 2.0;
        double r36687 = r36685 / r36686;
        double r36688 = y;
        double r36689 = cos(r36688);
        double r36690 = r36687 * r36689;
        double r36691 = r36682 - r36684;
        double r36692 = r36691 / r36686;
        double r36693 = sin(r36688);
        double r36694 = r36692 * r36693;
        double r36695 = /* ERROR: no complex support in C */;
        double r36696 = /* ERROR: no complex support in C */;
        return r36696;
}
↓
double f(double x, double y) {
        double r36697 = x;
        double r36698 = exp(r36697);
        double r36699 = -r36697;
        double r36700 = exp(r36699);
        double r36701 = r36698 + r36700;
        double r36702 = 2.0;
        double r36703 = r36701 / r36702;
        double r36704 = y;
        double r36705 = cos(r36704);
        double r36706 = r36703 * r36705;
        double r36707 = r36698 - r36700;
        double r36708 = r36707 / r36702;
        double r36709 = sin(r36704);
        double r36710 = r36708 * r36709;
        double r36711 = /* ERROR: no complex support in C */;
        double r36712 = /* ERROR: no complex support in C */;
        return r36712;
}

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 2019303 
(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)))))