Average Error: 19.9 → 0.2
Time: 4.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right) \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right) \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r425958 = x;
        double r425959 = y;
        double r425960 = r425958 * r425959;
        double r425961 = r425958 + r425959;
        double r425962 = r425961 * r425961;
        double r425963 = 1.0;
        double r425964 = r425961 + r425963;
        double r425965 = r425962 * r425964;
        double r425966 = r425960 / r425965;
        return r425966;
}
↓
double f(double x, double y) {
        double r425967 = 1.0;
        double r425968 = x;
        double r425969 = y;
        double r425970 = r425968 + r425969;
        double r425971 = r425967 / r425970;
        double r425972 = r425968 / r425970;
        double r425973 = r425971 * r425972;
        double r425974 = 1.0;
        double r425975 = r425970 + r425974;
        double r425976 = r425969 / r425975;
        double r425977 = r425973 * r425976;
        return r425977;
}

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

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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))))