Average Error: 19.5 → 0.2
Time: 38.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1}{y}}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1}{y}}
double f(double x, double y) {
        double r16304851 = x;
        double r16304852 = y;
        double r16304853 = r16304851 * r16304852;
        double r16304854 = r16304851 + r16304852;
        double r16304855 = r16304854 * r16304854;
        double r16304856 = 1.0;
        double r16304857 = r16304854 + r16304856;
        double r16304858 = r16304855 * r16304857;
        double r16304859 = r16304853 / r16304858;
        return r16304859;
}
↓
double f(double x, double y) {
        double r16304860 = x;
        double r16304861 = y;
        double r16304862 = r16304861 + r16304860;
        double r16304863 = r16304860 / r16304862;
        double r16304864 = r16304863 / r16304862;
        double r16304865 = 1.0;
        double r16304866 = 1.0;
        double r16304867 = r16304862 + r16304866;
        double r16304868 = r16304867 / r16304861;
        double r16304869 = r16304865 / r16304868;
        double r16304870 = r16304864 * r16304869;
        return r16304870;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.1
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied times-frac8.1

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

  5. Applied associate-/r*0.2

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

  7. Applied clear-num0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))