\[\frac{x}{x \cdot x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\ \end{array}\]

\frac{x}{x \cdot x + 1}

↓

\begin{array}{l}
\mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\

\end{array}

double f(double x) {
        double r53770 = x;
        double r53771 = r53770 * r53770;
        double r53772 = 1.0;
        double r53773 = r53771 + r53772;
        double r53774 = r53770 / r53773;
        return r53774;
}

↓

double f(double x) {
        double r53775 = x;
        double r53776 = -14917877722.165144;
        bool r53777 = r53775 <= r53776;
        double r53778 = 764.6675213818143;
        bool r53779 = r53775 <= r53778;
        double r53780 = !r53779;
        bool r53781 = r53777 || r53780;
        double r53782 = 1.0;
        double r53783 = 1.0;
        double r53784 = 5.0;
        double r53785 = pow(r53775, r53784);
        double r53786 = r53783 / r53785;
        double r53787 = 3.0;
        double r53788 = pow(r53775, r53787);
        double r53789 = r53783 / r53788;
        double r53790 = r53786 - r53789;
        double r53791 = r53783 / r53775;
        double r53792 = fma(r53782, r53790, r53791);
        double r53793 = r53775 * r53775;
        double r53794 = r53793 + r53782;
        double r53795 = sqrt(r53794);
        double r53796 = r53775 / r53795;
        double r53797 = 1.5;
        double r53798 = pow(r53782, r53797);
        double r53799 = hypot(r53798, r53788);
        double r53800 = r53793 * r53793;
        double r53801 = r53782 * r53782;
        double r53802 = r53793 * r53782;
        double r53803 = r53801 - r53802;
        double r53804 = r53800 + r53803;
        double r53805 = sqrt(r53804);
        double r53806 = r53799 / r53805;
        double r53807 = r53796 / r53806;
        double r53808 = r53781 ? r53792 : r53807;
        return r53808;
}

Original	15.0
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -14917877722.165144 or 764.6675213818143 < x
1. Initial program 30.2
  \[\frac{x}{x \cdot x + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right) - 1 \cdot \frac{1}{{x}^{3}}}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)}\]
if -14917877722.165144 < x < 764.6675213818143
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.0
  \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]
4. Applied associate-/r*0.0
  \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
5. Using strategy rm
6. Applied flip3-+0.0
  \[\leadsto \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {1}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}\]
7. Applied sqrt-div0.0
  \[\leadsto \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\color{blue}{\frac{\sqrt{{\left(x \cdot x\right)}^{3} + {1}^{3}}}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}\]
8. Simplified0.0
  \[\leadsto \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\color{blue}{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\ \end{array}\]

Error

Target

Derivation

`if x < -14917877722.165144 or 764.6675213818143 < x`

`if -14917877722.165144 < x < 764.6675213818143`

Reproduce