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

↓

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

\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 r778407 = x;
        double r778408 = r778407 * r778407;
        double r778409 = y;
        double r778410 = 4.0;
        double r778411 = r778409 * r778410;
        double r778412 = r778411 * r778409;
        double r778413 = r778408 - r778412;
        double r778414 = r778408 + r778412;
        double r778415 = r778413 / r778414;
        return r778415;
}

↓

double f(double x, double y) {
        double r778416 = x;
        double r778417 = -1.1020932357029451e+154;
        bool r778418 = r778416 <= r778417;
        double r778419 = 1.0;
        double r778420 = -7.055703835942177e-134;
        bool r778421 = r778416 <= r778420;
        double r778422 = r778416 * r778416;
        double r778423 = y;
        double r778424 = 4.0;
        double r778425 = r778423 * r778424;
        double r778426 = r778425 * r778423;
        double r778427 = r778422 + r778426;
        double r778428 = r778427 / r778422;
        double r778429 = r778419 / r778428;
        double r778430 = r778426 / r778427;
        double r778431 = r778429 - r778430;
        double r778432 = 3.3095785565051797e-97;
        bool r778433 = r778416 <= r778432;
        double r778434 = -1.0;
        double r778435 = 2.70835173311075e+105;
        bool r778436 = r778416 <= r778435;
        double r778437 = r778436 ? r778431 : r778419;
        double r778438 = r778433 ? r778434 : r778437;
        double r778439 = r778421 ? r778431 : r778438;
        double r778440 = r778418 ? r778419 : r778439;
        return r778440;
}

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))))

Falling back on racket egraph because egg-herbie package not installed (more)

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