Average Error: 0.0 → 0.0
Time: 2.1m
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 r18689 = x;
        double r18690 = exp(r18689);
        double r18691 = -r18689;
        double r18692 = exp(r18691);
        double r18693 = r18690 + r18692;
        double r18694 = 2.0;
        double r18695 = r18693 / r18694;
        double r18696 = y;
        double r18697 = cos(r18696);
        double r18698 = r18695 * r18697;
        double r18699 = r18690 - r18692;
        double r18700 = r18699 / r18694;
        double r18701 = sin(r18696);
        double r18702 = r18700 * r18701;
        double r18703 = /* ERROR: no complex support in C */;
        double r18704 = /* ERROR: no complex support in C */;
        return r18704;
}
↓
double f(double x, double y) {
        double r18705 = x;
        double r18706 = exp(r18705);
        double r18707 = -r18705;
        double r18708 = exp(r18707);
        double r18709 = r18706 + r18708;
        double r18710 = 2.0;
        double r18711 = r18709 / r18710;
        double r18712 = y;
        double r18713 = cos(r18712);
        double r18714 = r18711 * r18713;
        return r18714;
}

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 2019200 
(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)))))