Average Error: 20.2 → 0.1
Time: 15.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{y + x} \cdot \frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{y + x} \cdot \frac{\frac{y}{\left(y + x\right) + 1}}{y + x}
double f(double x, double y) {
        double r18911120 = x;
        double r18911121 = y;
        double r18911122 = r18911120 * r18911121;
        double r18911123 = r18911120 + r18911121;
        double r18911124 = r18911123 * r18911123;
        double r18911125 = 1.0;
        double r18911126 = r18911123 + r18911125;
        double r18911127 = r18911124 * r18911126;
        double r18911128 = r18911122 / r18911127;
        return r18911128;
}
↓
double f(double x, double y) {
        double r18911129 = x;
        double r18911130 = y;
        double r18911131 = r18911130 + r18911129;
        double r18911132 = r18911129 / r18911131;
        double r18911133 = 1.0;
        double r18911134 = r18911131 + r18911133;
        double r18911135 = r18911130 / r18911134;
        double r18911136 = r18911135 / r18911131;
        double r18911137 = r18911132 * r18911136;
        return r18911137;
}

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-frac8.2

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 div-inv0.2

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

  8. Applied associate-*l*0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 
(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))))