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 r32852 = x;
        double r32853 = exp(r32852);
        double r32854 = -r32852;
        double r32855 = exp(r32854);
        double r32856 = r32853 + r32855;
        double r32857 = 2.0;
        double r32858 = r32856 / r32857;
        double r32859 = y;
        double r32860 = cos(r32859);
        double r32861 = r32858 * r32860;
        double r32862 = r32853 - r32855;
        double r32863 = r32862 / r32857;
        double r32864 = sin(r32859);
        double r32865 = r32863 * r32864;
        double r32866 = /* ERROR: no complex support in C */;
        double r32867 = /* ERROR: no complex support in C */;
        return r32867;
}
↓
double f(double x, double y) {
        double r32868 = x;
        double r32869 = exp(r32868);
        double r32870 = -r32868;
        double r32871 = exp(r32870);
        double r32872 = r32869 + r32871;
        double r32873 = 2.0;
        double r32874 = r32872 / r32873;
        double r32875 = y;
        double r32876 = cos(r32875);
        double r32877 = r32874 * r32876;
        return r32877;
}

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)))))