Average Error: 0.0 → 0.0
Time: 28.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}} + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}} + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}
double f(double x, double y) {
        double r1370735 = x;
        double r1370736 = exp(r1370735);
        double r1370737 = -r1370735;
        double r1370738 = exp(r1370737);
        double r1370739 = r1370736 + r1370738;
        double r1370740 = 2.0;
        double r1370741 = r1370739 / r1370740;
        double r1370742 = y;
        double r1370743 = cos(r1370742);
        double r1370744 = r1370741 * r1370743;
        double r1370745 = r1370736 - r1370738;
        double r1370746 = r1370745 / r1370740;
        double r1370747 = sin(r1370742);
        double r1370748 = r1370746 * r1370747;
        double r1370749 = /* ERROR: no complex support in C */;
        double r1370750 = /* ERROR: no complex support in C */;
        return r1370750;
}
↓
double f(double x, double y) {
        double r1370751 = y;
        double r1370752 = cos(r1370751);
        double r1370753 = x;
        double r1370754 = exp(r1370753);
        double r1370755 = sqrt(r1370754);
        double r1370756 = r1370752 * r1370755;
        double r1370757 = r1370756 * r1370755;
        double r1370758 = r1370752 / r1370755;
        double r1370759 = r1370758 / r1370755;
        double r1370760 = r1370757 + r1370759;
        double r1370761 = 2.0;
        double r1370762 = r1370760 / r1370761;
        return r1370762;
}

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{\frac{\cos y}{e^{x}} + \cos y \cdot \color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}{2}\]

  5. Applied associate-*r*0.0

    \[\leadsto \frac{\frac{\cos y}{e^{x}} + \color{blue}{\left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}}}}{2}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{\frac{\cos y}{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}} + \left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}}}{2}\]

  8. Applied associate-/r*0.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}} + \left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}}}{2}\]

  9. Final simplification0.0

    \[\leadsto \frac{\left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}} + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))