Average Error: 0.0 → 0.0
Time: 21.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r688232 = x;
        double r688233 = exp(r688232);
        double r688234 = -r688232;
        double r688235 = exp(r688234);
        double r688236 = r688233 + r688235;
        double r688237 = 2.0;
        double r688238 = r688236 / r688237;
        double r688239 = y;
        double r688240 = cos(r688239);
        double r688241 = r688238 * r688240;
        double r688242 = r688233 - r688235;
        double r688243 = r688242 / r688237;
        double r688244 = sin(r688239);
        double r688245 = r688243 * r688244;
        double r688246 = /* ERROR: no complex support in C */;
        double r688247 = /* ERROR: no complex support in C */;
        return r688247;
}
↓
double f(double x, double y) {
        double r688248 = x;
        double r688249 = exp(r688248);
        double r688250 = -r688248;
        double r688251 = exp(r688250);
        double r688252 = r688249 + r688251;
        double r688253 = 2.0;
        double r688254 = r688252 / r688253;
        double r688255 = y;
        double r688256 = cos(r688255);
        double r688257 = r688254 * r688256;
        double r688258 = r688249 - r688251;
        double r688259 = r688258 / r688253;
        double r688260 = sin(r688255);
        double r688261 = r688259 * r688260;
        double r688262 = /* ERROR: no complex support in C */;
        double r688263 = /* ERROR: no complex support in C */;
        return r688263;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019120 
(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)))))