Average Error: 0.0 → 0.0
Time: 4.5s
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 r26593 = x;
        double r26594 = exp(r26593);
        double r26595 = -r26593;
        double r26596 = exp(r26595);
        double r26597 = r26594 + r26596;
        double r26598 = 2.0;
        double r26599 = r26597 / r26598;
        double r26600 = y;
        double r26601 = cos(r26600);
        double r26602 = r26599 * r26601;
        double r26603 = r26594 - r26596;
        double r26604 = r26603 / r26598;
        double r26605 = sin(r26600);
        double r26606 = r26604 * r26605;
        double r26607 = /* ERROR: no complex support in C */;
        double r26608 = /* ERROR: no complex support in C */;
        return r26608;
}
↓
double f(double x, double y) {
        double r26609 = x;
        double r26610 = exp(r26609);
        double r26611 = -r26609;
        double r26612 = exp(r26611);
        double r26613 = r26610 + r26612;
        double r26614 = 2.0;
        double r26615 = r26613 / r26614;
        double r26616 = y;
        double r26617 = cos(r26616);
        double r26618 = r26615 * r26617;
        return r26618;
}

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 2020049 +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)))))