Average Error: 0.0 → 0.0
Time: 8.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]
\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} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}
double f(double x, double y) {
        double r360185 = x;
        double r360186 = exp(r360185);
        double r360187 = -r360185;
        double r360188 = exp(r360187);
        double r360189 = r360186 + r360188;
        double r360190 = 2.0;
        double r360191 = r360189 / r360190;
        double r360192 = y;
        double r360193 = cos(r360192);
        double r360194 = r360191 * r360193;
        double r360195 = r360186 - r360188;
        double r360196 = r360195 / r360190;
        double r360197 = sin(r360192);
        double r360198 = r360196 * r360197;
        double r360199 = /* ERROR: no complex support in C */;
        double r360200 = /* ERROR: no complex support in C */;
        return r360200;
}
↓
double f(double x, double y) {
        double r360201 = x;
        double r360202 = exp(r360201);
        double r360203 = y;
        double r360204 = cos(r360203);
        double r360205 = r360202 * r360204;
        double r360206 = r360204 / r360202;
        double r360207 = r360205 + r360206;
        double r360208 = 2.0;
        double r360209 = r360207 / r360208;
        return r360209;
}

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))