Average Error: 0.0 → 0.0
Time: 17.8s
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 r32217 = x;
        double r32218 = exp(r32217);
        double r32219 = -r32217;
        double r32220 = exp(r32219);
        double r32221 = r32218 + r32220;
        double r32222 = 2.0;
        double r32223 = r32221 / r32222;
        double r32224 = y;
        double r32225 = cos(r32224);
        double r32226 = r32223 * r32225;
        double r32227 = r32218 - r32220;
        double r32228 = r32227 / r32222;
        double r32229 = sin(r32224);
        double r32230 = r32228 * r32229;
        double r32231 = /* ERROR: no complex support in C */;
        double r32232 = /* ERROR: no complex support in C */;
        return r32232;
}
↓
double f(double x, double y) {
        double r32233 = x;
        double r32234 = exp(r32233);
        double r32235 = -r32233;
        double r32236 = exp(r32235);
        double r32237 = r32234 + r32236;
        double r32238 = 2.0;
        double r32239 = r32237 / r32238;
        double r32240 = y;
        double r32241 = cos(r32240);
        double r32242 = r32239 * r32241;
        return r32242;
}

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 
(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)))))