Average Error: 19.5 → 0.2
Time: 17.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1}{y}}\]
\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}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1}{y}}
double f(double x, double y) {
        double r20314709 = x;
        double r20314710 = y;
        double r20314711 = r20314709 * r20314710;
        double r20314712 = r20314709 + r20314710;
        double r20314713 = r20314712 * r20314712;
        double r20314714 = 1.0;
        double r20314715 = r20314712 + r20314714;
        double r20314716 = r20314713 * r20314715;
        double r20314717 = r20314711 / r20314716;
        return r20314717;
}
↓
double f(double x, double y) {
        double r20314718 = x;
        double r20314719 = y;
        double r20314720 = r20314719 + r20314718;
        double r20314721 = r20314718 / r20314720;
        double r20314722 = r20314721 / r20314720;
        double r20314723 = 1.0;
        double r20314724 = 1.0;
        double r20314725 = r20314720 + r20314724;
        double r20314726 = r20314725 / r20314719;
        double r20314727 = r20314723 / r20314726;
        double r20314728 = r20314722 * r20314727;
        return r20314728;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
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 19.5

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 clear-num0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))