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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.10209323570294511 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -7.05570383594217736 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \le 3.30957855650517974 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

\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 -1.10209323570294511 \cdot 10^{154}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -7.05570383594217736 \cdot 10^{-134}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\

\mathbf{elif}\;x \le 3.30957855650517974 \cdot 10^{-97}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\

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

\end{array}

double f(double x, double y) {
        double r531635 = x;
        double r531636 = r531635 * r531635;
        double r531637 = y;
        double r531638 = 4.0;
        double r531639 = r531637 * r531638;
        double r531640 = r531639 * r531637;
        double r531641 = r531636 - r531640;
        double r531642 = r531636 + r531640;
        double r531643 = r531641 / r531642;
        return r531643;
}

↓

double f(double x, double y) {
        double r531644 = x;
        double r531645 = -1.1020932357029451e+154;
        bool r531646 = r531644 <= r531645;
        double r531647 = 1.0;
        double r531648 = -7.055703835942177e-134;
        bool r531649 = r531644 <= r531648;
        double r531650 = r531644 * r531644;
        double r531651 = y;
        double r531652 = 4.0;
        double r531653 = r531651 * r531652;
        double r531654 = r531653 * r531651;
        double r531655 = r531650 + r531654;
        double r531656 = r531655 / r531650;
        double r531657 = r531647 / r531656;
        double r531658 = r531654 / r531655;
        double r531659 = r531657 - r531658;
        double r531660 = 3.3095785565051797e-97;
        bool r531661 = r531644 <= r531660;
        double r531662 = -1.0;
        double r531663 = 2.70835173311075e+105;
        bool r531664 = r531644 <= r531663;
        double r531665 = r531664 ? r531659 : r531647;
        double r531666 = r531661 ? r531662 : r531665;
        double r531667 = r531649 ? r531659 : r531666;
        double r531668 = r531646 ? r531647 : r531667;
        return r531668;
}

Original	32.4
Target	32.1
Herbie	12.2

Derivation

Split input into 3 regimes
if x < -1.1020932357029451e+154 or 2.70835173311075e+105 < x
1. Initial program 57.5
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Taylor expanded around inf 9.0
  \[\leadsto \color{blue}{1}\]
if -1.1020932357029451e+154 < x < -7.055703835942177e-134 or 3.3095785565051797e-97 < x < 2.70835173311075e+105
1. Initial program 16.4
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Using strategy rm
3. Applied div-sub16.4
  \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\]
4. Using strategy rm
5. Applied clear-num16.4
  \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
if -7.055703835942177e-134 < x < 3.3095785565051797e-97
1. Initial program 28.8
  \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
2. Taylor expanded around 0 10.0
  \[\leadsto \color{blue}{-1}\]
Recombined 3 regimes into one program.
Final simplification12.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.10209323570294511 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -7.05570383594217736 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \le 3.30957855650517974 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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

Try it out

Target

Derivation

`if x < -1.1020932357029451e+154 or 2.70835173311075e+105 < x`

`if -1.1020932357029451e+154 < x < -7.055703835942177e-134 or 3.3095785565051797e-97 < x < 2.70835173311075e+105`

`if -7.055703835942177e-134 < x < 3.3095785565051797e-97`

Reproduce

In
Out