\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.6533496348093263 \cdot 10^{136}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -8.5079988842821871 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;x \le 5.33120441752882848 \cdot 10^{-100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.6140055087642122 \cdot 10^{150}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.6533496348093263 \cdot 10^{136}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -8.5079988842821871 \cdot 10^{-92}:\\
\;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\

\mathbf{elif}\;x \le 5.33120441752882848 \cdot 10^{-100}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 1.6140055087642122 \cdot 10^{150}:\\
\;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}

double f(double x, double y) {
        double r637712 = x;
        double r637713 = r637712 * r637712;
        double r637714 = y;
        double r637715 = 4.0;
        double r637716 = r637714 * r637715;
        double r637717 = r637716 * r637714;
        double r637718 = r637713 - r637717;
        double r637719 = r637713 + r637717;
        double r637720 = r637718 / r637719;
        return r637720;
}

↓

double f(double x, double y) {
        double r637721 = x;
        double r637722 = -1.6533496348093263e+136;
        bool r637723 = r637721 <= r637722;
        double r637724 = 1.0;
        double r637725 = -8.507998884282187e-92;
        bool r637726 = r637721 <= r637725;
        double r637727 = y;
        double r637728 = 4.0;
        double r637729 = r637727 * r637728;
        double r637730 = r637729 * r637727;
        double r637731 = fma(r637721, r637721, r637730);
        double r637732 = r637731 / r637721;
        double r637733 = r637721 / r637732;
        double r637734 = r637731 / r637727;
        double r637735 = r637729 / r637734;
        double r637736 = r637733 - r637735;
        double r637737 = 5.331204417528828e-100;
        bool r637738 = r637721 <= r637737;
        double r637739 = -1.0;
        double r637740 = 1.6140055087642122e+150;
        bool r637741 = r637721 <= r637740;
        double r637742 = r637741 ? r637736 : r637724;
        double r637743 = r637738 ? r637739 : r637742;
        double r637744 = r637726 ? r637736 : r637743;
        double r637745 = r637723 ? r637724 : r637744;
        return r637745;
}

Original	31.3
Target	31.0
Herbie	12.3

Derivation

Split input into 3 regimes
if x < -1.6533496348093263e+136 or 1.6140055087642122e+150 < x
1. Initial program 60.3
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Taylor expanded around inf 8.8
  \[\leadsto \color{blue}{1}\]
if -1.6533496348093263e+136 < x < -8.507998884282187e-92 or 5.331204417528828e-100 < x < 1.6140055087642122e+150
1. Initial program 15.6
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Using strategy rm
3. Applied div-sub15.5
  \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\]
4. Simplified15.6
  \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
5. Simplified15.1
  \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \color{blue}{\frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}}\]
if -8.507998884282187e-92 < x < 5.331204417528828e-100
1. Initial program 27.1
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Taylor expanded around 0 11.6
  \[\leadsto \color{blue}{-1}\]
Recombined 3 regimes into one program.
Final simplification12.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6533496348093263 \cdot 10^{136}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -8.5079988842821871 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;x \le 5.33120441752882848 \cdot 10^{-100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.6140055087642122 \cdot 10^{150}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))

Error

Target

Derivation

`if x < -1.6533496348093263e+136 or 1.6140055087642122e+150 < x`

`if -1.6533496348093263e+136 < x < -8.507998884282187e-92 or 5.331204417528828e-100 < x < 1.6140055087642122e+150`

`if -8.507998884282187e-92 < x < 5.331204417528828e-100`

Reproduce