Average Error: 0.0 → 0.0
Time: 13.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\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 r32753 = x;
        double r32754 = exp(r32753);
        double r32755 = -r32753;
        double r32756 = exp(r32755);
        double r32757 = r32754 + r32756;
        double r32758 = 2.0;
        double r32759 = r32757 / r32758;
        double r32760 = y;
        double r32761 = cos(r32760);
        double r32762 = r32759 * r32761;
        double r32763 = r32754 - r32756;
        double r32764 = r32763 / r32758;
        double r32765 = sin(r32760);
        double r32766 = r32764 * r32765;
        double r32767 = /* ERROR: no complex support in C */;
        double r32768 = /* ERROR: no complex support in C */;
        return r32768;
}
↓
double f(double x, double y) {
        double r32769 = x;
        double r32770 = exp(r32769);
        double r32771 = -r32769;
        double r32772 = exp(r32771);
        double r32773 = r32770 + r32772;
        double r32774 = 2.0;
        double r32775 = r32773 / r32774;
        double r32776 = y;
        double r32777 = cos(r32776);
        double r32778 = r32775 * r32777;
        return r32778;
}

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 2019212 
(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)))))