Average Error: 0.0 → 0.0
Time: 17.6s
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 r32653 = x;
        double r32654 = exp(r32653);
        double r32655 = -r32653;
        double r32656 = exp(r32655);
        double r32657 = r32654 + r32656;
        double r32658 = 2.0;
        double r32659 = r32657 / r32658;
        double r32660 = y;
        double r32661 = cos(r32660);
        double r32662 = r32659 * r32661;
        double r32663 = r32654 - r32656;
        double r32664 = r32663 / r32658;
        double r32665 = sin(r32660);
        double r32666 = r32664 * r32665;
        double r32667 = /* ERROR: no complex support in C */;
        double r32668 = /* ERROR: no complex support in C */;
        return r32668;
}
↓
double f(double x, double y) {
        double r32669 = x;
        double r32670 = exp(r32669);
        double r32671 = -r32669;
        double r32672 = exp(r32671);
        double r32673 = r32670 + r32672;
        double r32674 = 2.0;
        double r32675 = r32673 / r32674;
        double r32676 = y;
        double r32677 = cos(r32676);
        double r32678 = r32675 * r32677;
        return r32678;
}

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 2019326 +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)))))