Average Error: 0.0 → 0.0
Time: 27.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\left(\cos y \cdot \sqrt{e^{x}}\right) \cdot \sqrt{e^{x}} + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]
\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 r1370740 = x;
        double r1370741 = exp(r1370740);
        double r1370742 = -r1370740;
        double r1370743 = exp(r1370742);
        double r1370744 = r1370741 + r1370743;
        double r1370745 = 2.0;
        double r1370746 = r1370744 / r1370745;
        double r1370747 = y;
        double r1370748 = cos(r1370747);
        double r1370749 = r1370746 * r1370748;
        double r1370750 = r1370741 - r1370743;
        double r1370751 = r1370750 / r1370745;
        double r1370752 = sin(r1370747);
        double r1370753 = r1370751 * r1370752;
        double r1370754 = /* ERROR: no complex support in C */;
        double r1370755 = /* ERROR: no complex support in C */;
        return r1370755;
}
↓
double f(double x, double y) {
        double r1370756 = y;
        double r1370757 = cos(r1370756);
        double r1370758 = x;
        double r1370759 = exp(r1370758);
        double r1370760 = sqrt(r1370759);
        double r1370761 = r1370757 * r1370760;
        double r1370762 = r1370761 * r1370760;
        double r1370763 = r1370757 / r1370760;
        double r1370764 = r1370763 / r1370760;
        double r1370765 = r1370762 + r1370764;
        double r1370766 = 2.0;
        double r1370767 = r1370765 / r1370766;
        return r1370767;
}

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)))))