Average Error: 0.0 → 0.0
Time: 8.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]
\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} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}
double f(double x, double y) {
        double r1136641 = x;
        double r1136642 = exp(r1136641);
        double r1136643 = -r1136641;
        double r1136644 = exp(r1136643);
        double r1136645 = r1136642 + r1136644;
        double r1136646 = 2.0;
        double r1136647 = r1136645 / r1136646;
        double r1136648 = y;
        double r1136649 = cos(r1136648);
        double r1136650 = r1136647 * r1136649;
        double r1136651 = r1136642 - r1136644;
        double r1136652 = r1136651 / r1136646;
        double r1136653 = sin(r1136648);
        double r1136654 = r1136652 * r1136653;
        double r1136655 = /* ERROR: no complex support in C */;
        double r1136656 = /* ERROR: no complex support in C */;
        return r1136656;
}
↓
double f(double x, double y) {
        double r1136657 = x;
        double r1136658 = exp(r1136657);
        double r1136659 = y;
        double r1136660 = cos(r1136659);
        double r1136661 = r1136658 * r1136660;
        double r1136662 = r1136660 / r1136658;
        double r1136663 = r1136661 + r1136662;
        double r1136664 = 2.0;
        double r1136665 = r1136663 / r1136664;
        return r1136665;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))