Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\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 r28156 = x;
        double r28157 = exp(r28156);
        double r28158 = -r28156;
        double r28159 = exp(r28158);
        double r28160 = r28157 + r28159;
        double r28161 = 2.0;
        double r28162 = r28160 / r28161;
        double r28163 = y;
        double r28164 = cos(r28163);
        double r28165 = r28162 * r28164;
        double r28166 = r28157 - r28159;
        double r28167 = r28166 / r28161;
        double r28168 = sin(r28163);
        double r28169 = r28167 * r28168;
        double r28170 = /* ERROR: no complex support in C */;
        double r28171 = /* ERROR: no complex support in C */;
        return r28171;
}
↓
double f(double x, double y) {
        double r28172 = x;
        double r28173 = exp(r28172);
        double r28174 = -r28172;
        double r28175 = exp(r28174);
        double r28176 = r28173 + r28175;
        double r28177 = 2.0;
        double r28178 = r28176 / r28177;
        double r28179 = y;
        double r28180 = cos(r28179);
        double r28181 = r28178 * r28180;
        return r28181;
}

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 2019362 +o rules:numerics
(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)))))