Average Error: 0.0 → 0.0
Time: 6.0s
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 r41175 = x;
        double r41176 = exp(r41175);
        double r41177 = -r41175;
        double r41178 = exp(r41177);
        double r41179 = r41176 + r41178;
        double r41180 = 2.0;
        double r41181 = r41179 / r41180;
        double r41182 = y;
        double r41183 = cos(r41182);
        double r41184 = r41181 * r41183;
        double r41185 = r41176 - r41178;
        double r41186 = r41185 / r41180;
        double r41187 = sin(r41182);
        double r41188 = r41186 * r41187;
        double r41189 = /* ERROR: no complex support in C */;
        double r41190 = /* ERROR: no complex support in C */;
        return r41190;
}
↓
double f(double x, double y) {
        double r41191 = x;
        double r41192 = exp(r41191);
        double r41193 = -r41191;
        double r41194 = exp(r41193);
        double r41195 = r41192 + r41194;
        double r41196 = 2.0;
        double r41197 = r41195 / r41196;
        double r41198 = y;
        double r41199 = cos(r41198);
        double r41200 = r41197 * r41199;
        return r41200;
}

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 2019353 +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)))))