Average Error: 20.2 → 0.4
Time: 16.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{y + x}}{\frac{y + x}{\frac{y}{\left(y + x\right) + 1}}}\]
\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}{y + x}}{\frac{y + x}{\frac{y}{\left(y + x\right) + 1}}}
double f(double x, double y) {
        double r24987201 = x;
        double r24987202 = y;
        double r24987203 = r24987201 * r24987202;
        double r24987204 = r24987201 + r24987202;
        double r24987205 = r24987204 * r24987204;
        double r24987206 = 1.0;
        double r24987207 = r24987204 + r24987206;
        double r24987208 = r24987205 * r24987207;
        double r24987209 = r24987203 / r24987208;
        return r24987209;
}
↓
double f(double x, double y) {
        double r24987210 = x;
        double r24987211 = y;
        double r24987212 = r24987211 + r24987210;
        double r24987213 = r24987210 / r24987212;
        double r24987214 = 1.0;
        double r24987215 = r24987212 + r24987214;
        double r24987216 = r24987211 / r24987215;
        double r24987217 = r24987212 / r24987216;
        double r24987218 = r24987213 / r24987217;
        return r24987218;
}

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
Target0.2
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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-frac7.8

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 associate-/l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 +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))))