Average Error: 0.0 → 0.0
Time: 16.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}
double f(double x, double y) {
        double r785602 = x;
        double r785603 = exp(r785602);
        double r785604 = -r785602;
        double r785605 = exp(r785604);
        double r785606 = r785603 + r785605;
        double r785607 = 2.0;
        double r785608 = r785606 / r785607;
        double r785609 = y;
        double r785610 = cos(r785609);
        double r785611 = r785608 * r785610;
        double r785612 = r785603 - r785605;
        double r785613 = r785612 / r785607;
        double r785614 = sin(r785609);
        double r785615 = r785613 * r785614;
        double r785616 = /* ERROR: no complex support in C */;
        double r785617 = /* ERROR: no complex support in C */;
        return r785617;
}
↓
double f(double x, double y) {
        double r785618 = y;
        double r785619 = cos(r785618);
        double r785620 = x;
        double r785621 = exp(r785620);
        double r785622 = r785619 / r785621;
        double r785623 = r785621 * r785619;
        double r785624 = r785622 + r785623;
        double r785625 = 0.5;
        double r785626 = r785624 * r785625;
        return r785626;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}\right)}\]

  3. Final simplification0.0

    \[\leadsto \left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019130 
(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)))))