Average Error: 19.9 → 0.1
Time: 10.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{y + x} \cdot y}{y + x}}{\left(x + y\right) + 1}\]
\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{\frac{x}{y + x} \cdot y}{y + x}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r1175230 = x;
        double r1175231 = y;
        double r1175232 = r1175230 * r1175231;
        double r1175233 = r1175230 + r1175231;
        double r1175234 = r1175233 * r1175233;
        double r1175235 = 1.0;
        double r1175236 = r1175233 + r1175235;
        double r1175237 = r1175234 * r1175236;
        double r1175238 = r1175232 / r1175237;
        return r1175238;
}
↓
double f(double x, double y) {
        double r1175239 = x;
        double r1175240 = y;
        double r1175241 = r1175240 + r1175239;
        double r1175242 = r1175239 / r1175241;
        double r1175243 = r1175242 * r1175240;
        double r1175244 = r1175243 / r1175241;
        double r1175245 = r1175239 + r1175240;
        double r1175246 = 1.0;
        double r1175247 = r1175245 + r1175246;
        double r1175248 = r1175244 / r1175247;
        return r1175248;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
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 19.9

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 *-un-lft-identity8.0

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

  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 *-un-lft-identity0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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))))