Average Error: 19.8 → 0.2
Time: 15.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{1}{\frac{\left(x + y\right) + 1}{y}}}{x + y}\]
\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{1}{\frac{\left(x + y\right) + 1}{y}}}{x + y}
double f(double x, double y) {
        double r593672 = x;
        double r593673 = y;
        double r593674 = r593672 * r593673;
        double r593675 = r593672 + r593673;
        double r593676 = r593675 * r593675;
        double r593677 = 1.0;
        double r593678 = r593675 + r593677;
        double r593679 = r593676 * r593678;
        double r593680 = r593674 / r593679;
        return r593680;
}
↓
double f(double x, double y) {
        double r593681 = x;
        double r593682 = y;
        double r593683 = r593681 + r593682;
        double r593684 = r593681 / r593683;
        double r593685 = 1.0;
        double r593686 = 1.0;
        double r593687 = r593683 + r593686;
        double r593688 = r593687 / r593682;
        double r593689 = r593685 / r593688;
        double r593690 = r593684 * r593689;
        double r593691 = r593690 / r593683;
        return r593691;
}

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-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. Using strategy rm

  9. Applied clear-num0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))))