Average Error: 20.3 → 0.1
Time: 16.8s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r278958 = x;
        double r278959 = y;
        double r278960 = r278958 * r278959;
        double r278961 = r278958 + r278959;
        double r278962 = r278961 * r278961;
        double r278963 = 1.0;
        double r278964 = r278961 + r278963;
        double r278965 = r278962 * r278964;
        double r278966 = r278960 / r278965;
        return r278966;
}
↓
double f(double x, double y) {
        double r278967 = x;
        double r278968 = y;
        double r278969 = r278967 + r278968;
        double r278970 = r278967 / r278969;
        double r278971 = 1.0;
        double r278972 = r278969 + r278971;
        double r278973 = r278968 / r278972;
        double r278974 = r278970 * r278973;
        double r278975 = r278974 / r278969;
        return r278975;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
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 20.3

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

  3. Applied times-frac8.0

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 associate-*l/0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019198 +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))))