Average Error: 0.0 → 0.0
Time: 25.5s
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 r800033 = x;
        double r800034 = exp(r800033);
        double r800035 = -r800033;
        double r800036 = exp(r800035);
        double r800037 = r800034 + r800036;
        double r800038 = 2.0;
        double r800039 = r800037 / r800038;
        double r800040 = y;
        double r800041 = cos(r800040);
        double r800042 = r800039 * r800041;
        double r800043 = r800034 - r800036;
        double r800044 = r800043 / r800038;
        double r800045 = sin(r800040);
        double r800046 = r800044 * r800045;
        double r800047 = /* ERROR: no complex support in C */;
        double r800048 = /* ERROR: no complex support in C */;
        return r800048;
}
↓
double f(double x, double y) {
        double r800049 = x;
        double r800050 = exp(r800049);
        double r800051 = -r800049;
        double r800052 = exp(r800051);
        double r800053 = r800050 + r800052;
        double r800054 = 2.0;
        double r800055 = r800053 / r800054;
        double r800056 = y;
        double r800057 = cos(r800056);
        double r800058 = r800055 * r800057;
        double r800059 = r800050 - r800052;
        double r800060 = r800059 / r800054;
        double r800061 = sin(r800056);
        double r800062 = r800060 * r800061;
        double r800063 = /* ERROR: no complex support in C */;
        double r800064 = /* ERROR: no complex support in C */;
        return r800064;
}

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 2019151 
(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)))))