Average Error: 19.6 → 0.1
Time: 4.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}\]
\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{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r523040 = x;
        double r523041 = y;
        double r523042 = r523040 * r523041;
        double r523043 = r523040 + r523041;
        double r523044 = r523043 * r523043;
        double r523045 = 1.0;
        double r523046 = r523043 + r523045;
        double r523047 = r523044 * r523046;
        double r523048 = r523042 / r523047;
        return r523048;
}
↓
double f(double x, double y) {
        double r523049 = x;
        double r523050 = y;
        double r523051 = r523049 + r523050;
        double r523052 = r523049 / r523051;
        double r523053 = r523050 / r523051;
        double r523054 = r523052 * r523053;
        double r523055 = 1.0;
        double r523056 = r523051 + r523055;
        double r523057 = r523054 / r523056;
        return r523057;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
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 19.6

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

  3. Applied times-frac7.9

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*r/0.2

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

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 +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))))