Average Error: 19.7 → 0.5
Time: 6.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r389881 = x;
        double r389882 = y;
        double r389883 = r389881 * r389882;
        double r389884 = r389881 + r389882;
        double r389885 = r389884 * r389884;
        double r389886 = 1.0;
        double r389887 = r389884 + r389886;
        double r389888 = r389885 * r389887;
        double r389889 = r389883 / r389888;
        return r389889;
}
↓
double f(double x, double y) {
        double r389890 = 1.0;
        double r389891 = x;
        double r389892 = y;
        double r389893 = r389891 + r389892;
        double r389894 = r389891 / r389893;
        double r389895 = r389893 / r389894;
        double r389896 = r389890 / r389895;
        double r389897 = 1.0;
        double r389898 = r389893 + r389897;
        double r389899 = r389892 / r389898;
        double r389900 = r389896 * r389899;
        return r389900;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.1
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied times-frac7.5

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-num0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))