Average Error: 0.0 → 0.0
Time: 11.7s
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 r252220 = x;
        double r252221 = exp(r252220);
        double r252222 = -r252220;
        double r252223 = exp(r252222);
        double r252224 = r252221 + r252223;
        double r252225 = 2.0;
        double r252226 = r252224 / r252225;
        double r252227 = y;
        double r252228 = cos(r252227);
        double r252229 = r252226 * r252228;
        double r252230 = r252221 - r252223;
        double r252231 = r252230 / r252225;
        double r252232 = sin(r252227);
        double r252233 = r252231 * r252232;
        double r252234 = /* ERROR: no complex support in C */;
        double r252235 = /* ERROR: no complex support in C */;
        return r252235;
}
↓
double f(double x, double y) {
        double r252236 = x;
        double r252237 = exp(r252236);
        double r252238 = y;
        double r252239 = cos(r252238);
        double r252240 = r252237 * r252239;
        double r252241 = r252239 / r252237;
        double r252242 = r252240 + r252241;
        double r252243 = 2.0;
        double r252244 = r252242 / r252243;
        return r252244;
}

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 2019125 
(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)))))