Average Error: 32.4 → 12.2
Time: 4.6s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.10209323570294511 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -7.05570383594217736 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\ \mathbf{elif}\;x \le 3.30957855650517974 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\

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

\mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\
\;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\

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

\end{array}
double f(double x, double y) {
        double r625863 = x;
        double r625864 = r625863 * r625863;
        double r625865 = y;
        double r625866 = 4.0;
        double r625867 = r625865 * r625866;
        double r625868 = r625867 * r625865;
        double r625869 = r625864 - r625868;
        double r625870 = r625864 + r625868;
        double r625871 = r625869 / r625870;
        return r625871;
}
↓
double f(double x, double y) {
        double r625872 = x;
        double r625873 = -1.1020932357029451e+154;
        bool r625874 = r625872 <= r625873;
        double r625875 = 1.0;
        double r625876 = -7.055703835942177e-134;
        bool r625877 = r625872 <= r625876;
        double r625878 = 4.0;
        double r625879 = y;
        double r625880 = 2.0;
        double r625881 = pow(r625879, r625880);
        double r625882 = r625878 * r625881;
        double r625883 = fma(r625872, r625872, r625882);
        double r625884 = -r625882;
        double r625885 = fma(r625872, r625872, r625884);
        double r625886 = r625883 / r625885;
        double r625887 = 3.0;
        double r625888 = pow(r625886, r625887);
        double r625889 = cbrt(r625888);
        double r625890 = r625875 / r625889;
        double r625891 = 3.3095785565051797e-97;
        bool r625892 = r625872 <= r625891;
        double r625893 = -1.0;
        double r625894 = 2.70835173311075e+105;
        bool r625895 = r625872 <= r625894;
        double r625896 = r625895 ? r625890 : r625875;
        double r625897 = r625892 ? r625893 : r625896;
        double r625898 = r625877 ? r625890 : r625897;
        double r625899 = r625874 ? r625875 : r625898;
        return r625899;
}

Error

Bits error versus x

Bits error versus y

Target

Original32.4
Target32.1
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -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 clear-num16.4

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]

    4. Simplified16.4

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{{x}^{2} - 4 \cdot {y}^{2}}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube45.7

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{\color{blue}{\sqrt[3]{\left(\left({x}^{2} - 4 \cdot {y}^{2}\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)}}}}\]

    7. Applied add-cbrt-cube45.6

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right) \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left({x}^{2} - 4 \cdot {y}^{2}\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)}}}\]

    8. Applied cbrt-undiv45.6

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right) \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{\left(\left({x}^{2} - 4 \cdot {y}^{2}\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)\right) \cdot \left({x}^{2} - 4 \cdot {y}^{2}\right)}}}}\]

    9. Simplified16.4

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}}\]

    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

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

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.10209323570294511 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -7.05570383594217736 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\ \mathbf{elif}\;x \le 3.30957855650517974 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}{\mathsf{fma}\left(x, x, -4 \cdot {y}^{2}\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))