Average Error: 20.1 → 0.1
Time: 17.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{y + x} \cdot y}{1 + \left(y + x\right)}}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{\frac{x}{y + x} \cdot y}{1 + \left(y + x\right)}}{y + x}
double f(double x, double y) {
        double r16434176 = x;
        double r16434177 = y;
        double r16434178 = r16434176 * r16434177;
        double r16434179 = r16434176 + r16434177;
        double r16434180 = r16434179 * r16434179;
        double r16434181 = 1.0;
        double r16434182 = r16434179 + r16434181;
        double r16434183 = r16434180 * r16434182;
        double r16434184 = r16434178 / r16434183;
        return r16434184;
}
↓
double f(double x, double y) {
        double r16434185 = x;
        double r16434186 = y;
        double r16434187 = r16434186 + r16434185;
        double r16434188 = r16434185 / r16434187;
        double r16434189 = r16434188 * r16434186;
        double r16434190 = 1.0;
        double r16434191 = r16434190 + r16434187;
        double r16434192 = r16434189 / r16434191;
        double r16434193 = r16434192 / r16434187;
        return r16434193;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
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 20.1

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

  3. Applied times-frac8.1

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

  5. Applied *-un-lft-identity8.1

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  9. Applied associate-*l/7.3

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

  10. Applied associate-*r/7.3

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 +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))))