Average Error: 0.0 → 0.0
Time: 13.8s
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 r32758 = x;
        double r32759 = exp(r32758);
        double r32760 = -r32758;
        double r32761 = exp(r32760);
        double r32762 = r32759 + r32761;
        double r32763 = 2.0;
        double r32764 = r32762 / r32763;
        double r32765 = y;
        double r32766 = cos(r32765);
        double r32767 = r32764 * r32766;
        double r32768 = r32759 - r32761;
        double r32769 = r32768 / r32763;
        double r32770 = sin(r32765);
        double r32771 = r32769 * r32770;
        double r32772 = /* ERROR: no complex support in C */;
        double r32773 = /* ERROR: no complex support in C */;
        return r32773;
}
↓
double f(double x, double y) {
        double r32774 = x;
        double r32775 = exp(r32774);
        double r32776 = -r32774;
        double r32777 = exp(r32776);
        double r32778 = r32775 + r32777;
        double r32779 = 2.0;
        double r32780 = r32778 / r32779;
        double r32781 = y;
        double r32782 = cos(r32781);
        double r32783 = r32780 * r32782;
        return r32783;
}

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 2019208 
(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)))))