Average Error: 19.8 → 8.0
Time: 15.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \le 0.003245411500442616295480924293315183604136:\\ \;\;\;\;\frac{x}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \le 0.003245411500442616295480924293315183604136:\\
\;\;\;\;\frac{x}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double x, double y) {
        double r415918 = x;
        double r415919 = y;
        double r415920 = r415918 * r415919;
        double r415921 = r415918 + r415919;
        double r415922 = r415921 * r415921;
        double r415923 = 1.0;
        double r415924 = r415921 + r415923;
        double r415925 = r415922 * r415924;
        double r415926 = r415920 / r415925;
        return r415926;
}
↓
double f(double x, double y) {
        double r415927 = x;
        double r415928 = y;
        double r415929 = r415927 * r415928;
        double r415930 = r415927 + r415928;
        double r415931 = r415930 * r415930;
        double r415932 = 1.0;
        double r415933 = r415930 + r415932;
        double r415934 = r415931 * r415933;
        double r415935 = r415929 / r415934;
        double r415936 = 0.0032454115004426163;
        bool r415937 = r415935 <= r415936;
        double r415938 = r415927 * r415927;
        double r415939 = 2.0;
        double r415940 = r415939 * r415927;
        double r415941 = r415940 + r415928;
        double r415942 = r415928 * r415941;
        double r415943 = r415938 + r415942;
        double r415944 = r415927 / r415943;
        double r415945 = r415928 / r415933;
        double r415946 = r415944 * r415945;
        double r415947 = 0.0;
        double r415948 = r415937 ? r415946 : r415947;
        return r415948;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.1
Herbie8.0
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))) < 0.0032454115004426163

    1. Initial program 10.1

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]

    2. Taylor expanded around 0 10.1

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left({x}^{2} + \left({y}^{2} + 2 \cdot \left(x \cdot y\right)\right)\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

    3. Simplified10.1

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot x + y \cdot \left(2 \cdot x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)}\]
    4. Using strategy rm

    5. Applied times-frac2.0

      \[\leadsto \color{blue}{\frac{x}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]

    if 0.0032454115004426163 < (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))

    1. Initial program 63.4

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]

    2. Taylor expanded around inf 34.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \le 0.003245411500442616295480924293315183604136:\\ \;\;\;\;\frac{x}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1))))