Average Error: 0.0 → 0.1
Time: 5.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}{\sqrt[3]{2}} \cdot \cos y\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}{\sqrt[3]{2}} \cdot \cos y\right)
double f(double x, double y) {
        double r34668 = x;
        double r34669 = exp(r34668);
        double r34670 = -r34668;
        double r34671 = exp(r34670);
        double r34672 = r34669 + r34671;
        double r34673 = 2.0;
        double r34674 = r34672 / r34673;
        double r34675 = y;
        double r34676 = cos(r34675);
        double r34677 = r34674 * r34676;
        double r34678 = r34669 - r34671;
        double r34679 = r34678 / r34673;
        double r34680 = sin(r34675);
        double r34681 = r34679 * r34680;
        double r34682 = /* ERROR: no complex support in C */;
        double r34683 = /* ERROR: no complex support in C */;
        return r34683;
}
↓
double f(double x, double y) {
        double r34684 = x;
        double r34685 = exp(r34684);
        double r34686 = -r34684;
        double r34687 = exp(r34686);
        double r34688 = r34685 + r34687;
        double r34689 = cbrt(r34688);
        double r34690 = r34689 * r34689;
        double r34691 = 2.0;
        double r34692 = cbrt(r34691);
        double r34693 = r34692 * r34692;
        double r34694 = r34690 / r34693;
        double r34695 = 3.0;
        double r34696 = pow(r34685, r34695);
        double r34697 = pow(r34687, r34695);
        double r34698 = r34696 + r34697;
        double r34699 = cbrt(r34698);
        double r34700 = r34685 * r34685;
        double r34701 = r34687 * r34687;
        double r34702 = r34685 * r34687;
        double r34703 = r34701 - r34702;
        double r34704 = r34700 + r34703;
        double r34705 = cbrt(r34704);
        double r34706 = r34699 / r34705;
        double r34707 = r34706 / r34692;
        double r34708 = y;
        double r34709 = cos(r34708);
        double r34710 = r34707 * r34709;
        double r34711 = r34694 * r34710;
        return r34711;
}

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. Using strategy rm

  4. Applied add-cube-cbrt2.2

    \[\leadsto \frac{e^{x} + e^{-x}}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}} \cdot \cos y\]

  5. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} \cdot \cos y\]

  6. Applied times-frac0.0

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{\sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2}}\right)} \cdot \cos y\]

  7. Applied associate-*l*0.0

    \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2}} \cdot \cos y\right)}\]
  8. Using strategy rm

  9. Applied flip3-+0.1

    \[\leadsto \frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\sqrt[3]{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}}{\sqrt[3]{2}} \cdot \cos y\right)\]

  10. Applied cbrt-div0.1

    \[\leadsto \frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\color{blue}{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}}{\sqrt[3]{2}} \cdot \cos y\right)\]

  11. Final simplification0.1

    \[\leadsto \frac{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \left(\frac{\frac{\sqrt[3]{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}}{\sqrt[3]{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}{\sqrt[3]{2}} \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2020089 
(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)))))