Average Error: 19.2 → 0.1
Time: 32.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{y}{x + y} \cdot \frac{\frac{x}{x + y}}{1.0 + \left(x + y\right)}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{y}{x + y} \cdot \frac{\frac{x}{x + y}}{1.0 + \left(x + y\right)}
double f(double x, double y) {
        double r30654038 = x;
        double r30654039 = y;
        double r30654040 = r30654038 * r30654039;
        double r30654041 = r30654038 + r30654039;
        double r30654042 = r30654041 * r30654041;
        double r30654043 = 1.0;
        double r30654044 = r30654041 + r30654043;
        double r30654045 = r30654042 * r30654044;
        double r30654046 = r30654040 / r30654045;
        return r30654046;
}
↓
double f(double x, double y) {
        double r30654047 = y;
        double r30654048 = x;
        double r30654049 = r30654048 + r30654047;
        double r30654050 = r30654047 / r30654049;
        double r30654051 = r30654048 / r30654049;
        double r30654052 = 1.0;
        double r30654053 = r30654052 + r30654049;
        double r30654054 = r30654051 / r30654053;
        double r30654055 = r30654050 * r30654054;
        return r30654055;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.2
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 19.2

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity19.2

    \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(1 \cdot \left(x + y\right)\right)}\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]

  4. Applied associate-*r*19.2

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(\left(x + y\right) \cdot 1\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1.0\right)}\]

  5. Applied associate-*l*19.2

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot 1\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1.0\right)\right)}}\]

  6. Applied *-commutative19.2

    \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot 1\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1.0\right)\right)}\]

  7. Applied times-frac4.0

    \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1.0\right)}}\]

  8. Simplified4.0

    \[\leadsto \color{blue}{\frac{y}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]

  9. Simplified0.1

    \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1.0}}\]

  10. Final simplification0.1

    \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{x + y}}{1.0 + \left(x + y\right)}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

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