Average Error: 0.0 → 0.0
Time: 3.2s
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 r28976 = x;
        double r28977 = exp(r28976);
        double r28978 = -r28976;
        double r28979 = exp(r28978);
        double r28980 = r28977 + r28979;
        double r28981 = 2.0;
        double r28982 = r28980 / r28981;
        double r28983 = y;
        double r28984 = cos(r28983);
        double r28985 = r28982 * r28984;
        double r28986 = r28977 - r28979;
        double r28987 = r28986 / r28981;
        double r28988 = sin(r28983);
        double r28989 = r28987 * r28988;
        double r28990 = /* ERROR: no complex support in C */;
        double r28991 = /* ERROR: no complex support in C */;
        return r28991;
}
↓
double f(double x, double y) {
        double r28992 = x;
        double r28993 = exp(r28992);
        double r28994 = -r28992;
        double r28995 = exp(r28994);
        double r28996 = r28993 + r28995;
        double r28997 = 2.0;
        double r28998 = r28996 / r28997;
        double r28999 = y;
        double r29000 = cos(r28999);
        double r29001 = r28998 * r29000;
        return r29001;
}

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 2020018 
(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)))))