Average Error: 19.9 → 0.1
Time: 9.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}\]
\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{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r418616 = x;
        double r418617 = y;
        double r418618 = r418616 * r418617;
        double r418619 = r418616 + r418617;
        double r418620 = r418619 * r418619;
        double r418621 = 1.0;
        double r418622 = r418619 + r418621;
        double r418623 = r418620 * r418622;
        double r418624 = r418618 / r418623;
        return r418624;
}
↓
double f(double x, double y) {
        double r418625 = x;
        double r418626 = y;
        double r418627 = r418625 + r418626;
        double r418628 = r418625 / r418627;
        double r418629 = r418628 * r418626;
        double r418630 = 1.0;
        double r418631 = r418627 + r418630;
        double r418632 = r418629 / r418631;
        double r418633 = r418632 / r418627;
        return r418633;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
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 19.9

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 *-un-lft-identity8.0

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  9. Applied associate-*l/7.6

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

  10. Applied associate-*r/7.6

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))))