Average Error: 20.0 → 0.5
Time: 15.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{y}{\left(x + y\right) + 1} \cdot \frac{1}{\frac{x + y}{\frac{x}{x + y}}}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{y}{\left(x + y\right) + 1} \cdot \frac{1}{\frac{x + y}{\frac{x}{x + y}}}
double f(double x, double y) {
        double r16770189 = x;
        double r16770190 = y;
        double r16770191 = r16770189 * r16770190;
        double r16770192 = r16770189 + r16770190;
        double r16770193 = r16770192 * r16770192;
        double r16770194 = 1.0;
        double r16770195 = r16770192 + r16770194;
        double r16770196 = r16770193 * r16770195;
        double r16770197 = r16770191 / r16770196;
        return r16770197;
}
↓
double f(double x, double y) {
        double r16770198 = y;
        double r16770199 = x;
        double r16770200 = r16770199 + r16770198;
        double r16770201 = 1.0;
        double r16770202 = r16770200 + r16770201;
        double r16770203 = r16770198 / r16770202;
        double r16770204 = 1.0;
        double r16770205 = r16770199 / r16770200;
        double r16770206 = r16770200 / r16770205;
        double r16770207 = r16770204 / r16770206;
        double r16770208 = r16770203 * r16770207;
        return r16770208;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.0

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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-num0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +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))))