Average Error: 19.7 → 0.2
Time: 3.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r419044 = x;
        double r419045 = y;
        double r419046 = r419044 * r419045;
        double r419047 = r419044 + r419045;
        double r419048 = r419047 * r419047;
        double r419049 = 1.0;
        double r419050 = r419047 + r419049;
        double r419051 = r419048 * r419050;
        double r419052 = r419046 / r419051;
        return r419052;
}
↓
double f(double x, double y) {
        double r419053 = x;
        double r419054 = y;
        double r419055 = r419053 + r419054;
        double r419056 = r419053 / r419055;
        double r419057 = r419056 / r419055;
        double r419058 = 1.0;
        double r419059 = r419055 + r419058;
        double r419060 = r419054 / r419059;
        double r419061 = r419057 * r419060;
        return r419061;
}

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

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-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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020065 +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))))