Average Error: 0.0 → 0.0
Time: 4.8s
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 r37979 = x;
        double r37980 = exp(r37979);
        double r37981 = -r37979;
        double r37982 = exp(r37981);
        double r37983 = r37980 + r37982;
        double r37984 = 2.0;
        double r37985 = r37983 / r37984;
        double r37986 = y;
        double r37987 = cos(r37986);
        double r37988 = r37985 * r37987;
        double r37989 = r37980 - r37982;
        double r37990 = r37989 / r37984;
        double r37991 = sin(r37986);
        double r37992 = r37990 * r37991;
        double r37993 = /* ERROR: no complex support in C */;
        double r37994 = /* ERROR: no complex support in C */;
        return r37994;
}
↓
double f(double x, double y) {
        double r37995 = x;
        double r37996 = exp(r37995);
        double r37997 = -r37995;
        double r37998 = exp(r37997);
        double r37999 = r37996 + r37998;
        double r38000 = 2.0;
        double r38001 = r37999 / r38000;
        double r38002 = y;
        double r38003 = cos(r38002);
        double r38004 = r38001 * r38003;
        return r38004;
}

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 2020033 
(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)))))