Average Error: 0.0 → 0.0
Time: 24.2s
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 r43628 = x;
        double r43629 = exp(r43628);
        double r43630 = -r43628;
        double r43631 = exp(r43630);
        double r43632 = r43629 + r43631;
        double r43633 = 2.0;
        double r43634 = r43632 / r43633;
        double r43635 = y;
        double r43636 = cos(r43635);
        double r43637 = r43634 * r43636;
        double r43638 = r43629 - r43631;
        double r43639 = r43638 / r43633;
        double r43640 = sin(r43635);
        double r43641 = r43639 * r43640;
        double r43642 = /* ERROR: no complex support in C */;
        double r43643 = /* ERROR: no complex support in C */;
        return r43643;
}
↓
double f(double x, double y) {
        double r43644 = x;
        double r43645 = exp(r43644);
        double r43646 = -r43644;
        double r43647 = exp(r43646);
        double r43648 = r43645 + r43647;
        double r43649 = 2.0;
        double r43650 = r43648 / r43649;
        double r43651 = y;
        double r43652 = cos(r43651);
        double r43653 = r43650 * r43652;
        return r43653;
}

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 2019323 +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)))))