Average Error: 19.8 → 0.2
Time: 4.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}\]
\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{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r448568 = x;
        double r448569 = y;
        double r448570 = r448568 * r448569;
        double r448571 = r448568 + r448569;
        double r448572 = r448571 * r448571;
        double r448573 = 1.0;
        double r448574 = r448571 + r448573;
        double r448575 = r448572 * r448574;
        double r448576 = r448570 / r448575;
        return r448576;
}
↓
double f(double x, double y) {
        double r448577 = x;
        double r448578 = y;
        double r448579 = r448577 + r448578;
        double r448580 = r448577 / r448579;
        double r448581 = r448580 / r448579;
        double r448582 = r448581 * r448578;
        double r448583 = 1.0;
        double r448584 = r448579 + r448583;
        double r448585 = r448582 / r448584;
        return r448585;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
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 19.8

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

  3. Applied times-frac7.8

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*r/0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 
(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))))