Average Error: 19.4 → 0.1
Time: 4.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}
double f(double x, double y) {
        double r442970 = x;
        double r442971 = y;
        double r442972 = r442970 * r442971;
        double r442973 = r442970 + r442971;
        double r442974 = r442973 * r442973;
        double r442975 = 1.0;
        double r442976 = r442973 + r442975;
        double r442977 = r442974 * r442976;
        double r442978 = r442972 / r442977;
        return r442978;
}
↓
double f(double x, double y) {
        double r442979 = x;
        double r442980 = y;
        double r442981 = r442979 + r442980;
        double r442982 = 1.0;
        double r442983 = r442981 + r442982;
        double r442984 = r442980 / r442983;
        double r442985 = r442984 / r442981;
        double r442986 = r442979 * r442985;
        double r442987 = r442986 / r442981;
        return r442987;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.4

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

  3. Applied times-frac7.3

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

  5. Applied associate-/r*0.1

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

  7. Applied associate-*l/0.1

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

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 +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))))