Average Error: 0.0 → 0.0
Time: 31.7s
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 r585915 = x;
        double r585916 = exp(r585915);
        double r585917 = -r585915;
        double r585918 = exp(r585917);
        double r585919 = r585916 + r585918;
        double r585920 = 2.0;
        double r585921 = r585919 / r585920;
        double r585922 = y;
        double r585923 = cos(r585922);
        double r585924 = r585921 * r585923;
        double r585925 = r585916 - r585918;
        double r585926 = r585925 / r585920;
        double r585927 = sin(r585922);
        double r585928 = r585926 * r585927;
        double r585929 = /* ERROR: no complex support in C */;
        double r585930 = /* ERROR: no complex support in C */;
        return r585930;
}
↓
double f(double x, double y) {
        double r585931 = x;
        double r585932 = exp(r585931);
        double r585933 = y;
        double r585934 = cos(r585933);
        double r585935 = r585932 * r585934;
        double r585936 = r585934 / r585932;
        double r585937 = r585935 + r585936;
        double r585938 = 2.0;
        double r585939 = r585937 / r585938;
        return r585939;
}

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 2019168 
(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)))))