Average Error: 19.7 → 0.5
Time: 9.6s
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 r536976 = x;
        double r536977 = y;
        double r536978 = r536976 * r536977;
        double r536979 = r536976 + r536977;
        double r536980 = r536979 * r536979;
        double r536981 = 1.0;
        double r536982 = r536979 + r536981;
        double r536983 = r536980 * r536982;
        double r536984 = r536978 / r536983;
        return r536984;
}
↓
double f(double x, double y) {
        double r536985 = 1.0;
        double r536986 = x;
        double r536987 = y;
        double r536988 = r536986 + r536987;
        double r536989 = r536986 / r536988;
        double r536990 = r536988 / r536989;
        double r536991 = r536985 / r536990;
        double r536992 = 1.0;
        double r536993 = r536988 + r536992;
        double r536994 = r536987 / r536993;
        double r536995 = r536991 * r536994;
        return r536995;
}

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 *-un-lft-identity0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-/l*0.5

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

  11. 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 
(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))))