Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r19627 = x;
        double r19628 = exp(r19627);
        double r19629 = -r19627;
        double r19630 = exp(r19629);
        double r19631 = r19628 + r19630;
        double r19632 = 2.0;
        double r19633 = r19631 / r19632;
        double r19634 = y;
        double r19635 = cos(r19634);
        double r19636 = r19633 * r19635;
        double r19637 = r19628 - r19630;
        double r19638 = r19637 / r19632;
        double r19639 = sin(r19634);
        double r19640 = r19638 * r19639;
        double r19641 = /* ERROR: no complex support in C */;
        double r19642 = /* ERROR: no complex support in C */;
        return r19642;
}
↓
double f(double x, double y) {
        double r19643 = x;
        double r19644 = exp(r19643);
        double r19645 = -r19643;
        double r19646 = exp(r19645);
        double r19647 = r19644 + r19646;
        double r19648 = 2.0;
        double r19649 = r19647 / r19648;
        double r19650 = y;
        double r19651 = cos(r19650);
        double r19652 = r19649 * r19651;
        return r19652;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2} \cdot \cos y}\]

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} + e^{-x}}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(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)))))