Average Error: 31.1 → 13.3
Time: 8.2s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\
\;\;\;\;\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}\\

\mathbf{elif}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\
\;\;\;\;\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}\\

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

\end{array}
double f(double x, double y) {
        double r26499268 = x;
        double r26499269 = r26499268 * r26499268;
        double r26499270 = y;
        double r26499271 = 4.0;
        double r26499272 = r26499270 * r26499271;
        double r26499273 = r26499272 * r26499270;
        double r26499274 = r26499269 - r26499273;
        double r26499275 = r26499269 + r26499273;
        double r26499276 = r26499274 / r26499275;
        return r26499276;
}
↓
double f(double x, double y) {
        double r26499277 = x;
        double r26499278 = r26499277 * r26499277;
        double r26499279 = 2.4742170052004425e-201;
        bool r26499280 = r26499278 <= r26499279;
        double r26499281 = -1.0;
        double r26499282 = 2.3542361087629974e-102;
        bool r26499283 = r26499278 <= r26499282;
        double r26499284 = y;
        double r26499285 = 4.0;
        double r26499286 = r26499284 * r26499285;
        double r26499287 = r26499286 * r26499284;
        double r26499288 = r26499278 + r26499287;
        double r26499289 = r26499278 / r26499288;
        double r26499290 = r26499287 / r26499288;
        double r26499291 = r26499289 - r26499290;
        double r26499292 = 5.880877246270695e-09;
        bool r26499293 = r26499278 <= r26499292;
        double r26499294 = 5.6317960521962586e+278;
        bool r26499295 = r26499278 <= r26499294;
        double r26499296 = 1.0;
        double r26499297 = r26499295 ? r26499291 : r26499296;
        double r26499298 = r26499293 ? r26499281 : r26499297;
        double r26499299 = r26499283 ? r26499291 : r26499298;
        double r26499300 = r26499280 ? r26499281 : r26499299;
        return r26499300;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.1
Target30.8
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x x) < 2.4742170052004425e-201 or 2.3542361087629974e-102 < (* x x) < 5.880877246270695e-09

    1. Initial program 24.2

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

    2. Taylor expanded around 0 13.7

      \[\leadsto \color{blue}{-1}\]

    if 2.4742170052004425e-201 < (* x x) < 2.3542361087629974e-102 or 5.880877246270695e-09 < (* x x) < 5.6317960521962586e+278

    1. Initial program 16.1

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
    2. Using strategy rm

    3. Applied div-sub16.1

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

    if 5.6317960521962586e+278 < (* x x)

    1. Initial program 59.5

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

    2. Taylor expanded around inf 9.3

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))