Average Error: 0.0 → 0.0
Time: 18.9s
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 r1647823 = x;
        double r1647824 = exp(r1647823);
        double r1647825 = -r1647823;
        double r1647826 = exp(r1647825);
        double r1647827 = r1647824 + r1647826;
        double r1647828 = 2.0;
        double r1647829 = r1647827 / r1647828;
        double r1647830 = y;
        double r1647831 = cos(r1647830);
        double r1647832 = r1647829 * r1647831;
        double r1647833 = r1647824 - r1647826;
        double r1647834 = r1647833 / r1647828;
        double r1647835 = sin(r1647830);
        double r1647836 = r1647834 * r1647835;
        double r1647837 = /* ERROR: no complex support in C */;
        double r1647838 = /* ERROR: no complex support in C */;
        return r1647838;
}
↓
double f(double x, double y) {
        double r1647839 = x;
        double r1647840 = exp(r1647839);
        double r1647841 = y;
        double r1647842 = cos(r1647841);
        double r1647843 = r1647840 * r1647842;
        double r1647844 = r1647842 / r1647840;
        double r1647845 = r1647843 + r1647844;
        double r1647846 = 2.0;
        double r1647847 = r1647845 / r1647846;
        return r1647847;
}

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))