Average Error: 0.0 → 0.0
Time: 20.1s
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 r1172774 = x;
        double r1172775 = exp(r1172774);
        double r1172776 = -r1172774;
        double r1172777 = exp(r1172776);
        double r1172778 = r1172775 + r1172777;
        double r1172779 = 2.0;
        double r1172780 = r1172778 / r1172779;
        double r1172781 = y;
        double r1172782 = cos(r1172781);
        double r1172783 = r1172780 * r1172782;
        double r1172784 = r1172775 - r1172777;
        double r1172785 = r1172784 / r1172779;
        double r1172786 = sin(r1172781);
        double r1172787 = r1172785 * r1172786;
        double r1172788 = /* ERROR: no complex support in C */;
        double r1172789 = /* ERROR: no complex support in C */;
        return r1172789;
}
↓
double f(double x, double y) {
        double r1172790 = x;
        double r1172791 = exp(r1172790);
        double r1172792 = y;
        double r1172793 = cos(r1172792);
        double r1172794 = r1172791 * r1172793;
        double r1172795 = r1172793 / r1172791;
        double r1172796 = r1172794 + r1172795;
        double r1172797 = 2.0;
        double r1172798 = r1172796 / r1172797;
        return r1172798;
}

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 2019170 
(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)))))