Average Error: 0.0 → 0.0
Time: 3.6s
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 r24634 = x;
        double r24635 = exp(r24634);
        double r24636 = -r24634;
        double r24637 = exp(r24636);
        double r24638 = r24635 + r24637;
        double r24639 = 2.0;
        double r24640 = r24638 / r24639;
        double r24641 = y;
        double r24642 = cos(r24641);
        double r24643 = r24640 * r24642;
        double r24644 = r24635 - r24637;
        double r24645 = r24644 / r24639;
        double r24646 = sin(r24641);
        double r24647 = r24645 * r24646;
        double r24648 = /* ERROR: no complex support in C */;
        double r24649 = /* ERROR: no complex support in C */;
        return r24649;
}
↓
double f(double x, double y) {
        double r24650 = x;
        double r24651 = exp(r24650);
        double r24652 = -r24650;
        double r24653 = exp(r24652);
        double r24654 = r24651 + r24653;
        double r24655 = 2.0;
        double r24656 = r24654 / r24655;
        double r24657 = y;
        double r24658 = cos(r24657);
        double r24659 = r24656 * r24658;
        return r24659;
}

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 2020021 
(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)))))