\[0.0 \lt x \lt 1 \land y \lt 1\]

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.329253761175223739159446383274506990081 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}

↓

\begin{array}{l}
\mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.329253761175223739159446383274506990081 \cdot 10^{-158}:\\
\;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\

\mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\

\end{array}

double f(double x, double y) {
        double r5592129 = x;
        double r5592130 = y;
        double r5592131 = r5592129 - r5592130;
        double r5592132 = r5592129 + r5592130;
        double r5592133 = r5592131 * r5592132;
        double r5592134 = r5592129 * r5592129;
        double r5592135 = r5592130 * r5592130;
        double r5592136 = r5592134 + r5592135;
        double r5592137 = r5592133 / r5592136;
        return r5592137;
}

↓

double f(double x, double y) {
        double r5592138 = y;
        double r5592139 = -5.797856225877881e+150;
        bool r5592140 = r5592138 <= r5592139;
        double r5592141 = -1.0;
        double r5592142 = -1.3292537611752237e-158;
        bool r5592143 = r5592138 <= r5592142;
        double r5592144 = x;
        double r5592145 = r5592144 * r5592144;
        double r5592146 = r5592138 * r5592138;
        double r5592147 = r5592145 + r5592146;
        double r5592148 = sqrt(r5592147);
        double r5592149 = r5592145 / r5592148;
        double r5592150 = r5592138 / r5592148;
        double r5592151 = r5592150 * r5592138;
        double r5592152 = r5592149 - r5592151;
        double r5592153 = r5592152 / r5592148;
        double r5592154 = 7.961862811311691e-164;
        bool r5592155 = r5592138 <= r5592154;
        double r5592156 = 1.0;
        double r5592157 = r5592155 ? r5592156 : r5592153;
        double r5592158 = r5592143 ? r5592153 : r5592157;
        double r5592159 = r5592140 ? r5592141 : r5592158;
        return r5592159;
}

Original	20.1
Target	0.0
Herbie	5.0

Derivation

Split input into 3 regimes
if y < -5.797856225877881e+150
1. Initial program 62.3
  \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
2. Taylor expanded around 0 0
  \[\leadsto \color{blue}{-1}\]
if -5.797856225877881e+150 < y < -1.3292537611752237e-158 or 7.961862811311691e-164 < y
1. Initial program 0.1
  \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.1
  \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}}\]
4. Applied associate-/r*0.2
  \[\leadsto \color{blue}{\frac{\frac{\left(x - y\right) \cdot \left(x + y\right)}{\sqrt{x \cdot x + y \cdot y}}}{\sqrt{x \cdot x + y \cdot y}}}\]
5. Simplified0.2
  \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y \cdot y}{\sqrt{x \cdot x + y \cdot y}}}}{\sqrt{x \cdot x + y \cdot y}}\]
6. Using strategy rm
7. Applied *-un-lft-identity0.2
  \[\leadsto \frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y \cdot y}{\sqrt{\color{blue}{1 \cdot \left(x \cdot x + y \cdot y\right)}}}}{\sqrt{x \cdot x + y \cdot y}}\]
8. Applied sqrt-prod0.2
  \[\leadsto \frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y \cdot y}{\color{blue}{\sqrt{1} \cdot \sqrt{x \cdot x + y \cdot y}}}}{\sqrt{x \cdot x + y \cdot y}}\]
9. Applied times-frac0.4
  \[\leadsto \frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \color{blue}{\frac{y}{\sqrt{1}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}}}{\sqrt{x \cdot x + y \cdot y}}\]
10. Simplified0.4
  \[\leadsto \frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \color{blue}{y} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}}{\sqrt{x \cdot x + y \cdot y}}\]
if -1.3292537611752237e-158 < y < 7.961862811311691e-164
1. Initial program 29.2
  \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
2. Taylor expanded around inf 15.0
  \[\leadsto \color{blue}{1}\]
Recombined 3 regimes into one program.
Final simplification5.0
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.329253761175223739159446383274506990081 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot x}{\sqrt{x \cdot x + y \cdot y}} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Error

Try it out

Target

Derivation

`if y < -5.797856225877881e+150`

`if -5.797856225877881e+150 < y < -1.3292537611752237e-158 or 7.961862811311691e-164 < y`

`if -1.3292537611752237e-158 < y < 7.961862811311691e-164`

Reproduce

In
Out