Average Error: 20.2 → 0.1
Time: 4.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}\]
\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{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r387551 = x;
        double r387552 = y;
        double r387553 = r387551 * r387552;
        double r387554 = r387551 + r387552;
        double r387555 = r387554 * r387554;
        double r387556 = 1.0;
        double r387557 = r387554 + r387556;
        double r387558 = r387555 * r387557;
        double r387559 = r387553 / r387558;
        return r387559;
}
↓
double f(double x, double y) {
        double r387560 = y;
        double r387561 = x;
        double r387562 = r387561 + r387560;
        double r387563 = r387561 / r387562;
        double r387564 = r387560 * r387563;
        double r387565 = r387564 / r387562;
        double r387566 = 1.0;
        double r387567 = r387562 + r387566;
        double r387568 = r387565 / r387567;
        return r387568;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
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 20.2

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

  3. Applied times-frac8.3

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

  5. Applied associate-/r*0.2

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

  7. Applied expm1-log1p-u0.2

    \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
  8. Using strategy rm

  9. Applied associate-*r/0.2

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)}{x + y} \cdot y}{\left(x + y\right) + 1}}\]

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 +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))))