Average Error: 19.3 → 0.1
Time: 18.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{\left(y + x\right) + 1.0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y + x}\right)\right)}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{\frac{y}{\left(y + x\right) + 1.0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y + x}\right)\right)}{y + x}
double f(double x, double y) {
        double r21936874 = x;
        double r21936875 = y;
        double r21936876 = r21936874 * r21936875;
        double r21936877 = r21936874 + r21936875;
        double r21936878 = r21936877 * r21936877;
        double r21936879 = 1.0;
        double r21936880 = r21936877 + r21936879;
        double r21936881 = r21936878 * r21936880;
        double r21936882 = r21936876 / r21936881;
        return r21936882;
}
↓
double f(double x, double y) {
        double r21936883 = y;
        double r21936884 = x;
        double r21936885 = r21936883 + r21936884;
        double r21936886 = 1.0;
        double r21936887 = r21936885 + r21936886;
        double r21936888 = r21936883 / r21936887;
        double r21936889 = r21936884 / r21936885;
        double r21936890 = log1p(r21936889);
        double r21936891 = expm1(r21936890);
        double r21936892 = r21936888 * r21936891;
        double r21936893 = r21936892 / r21936885;
        return r21936893;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original19.3
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 19.3

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac7.6

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(x + y\right) + 1.0}\]
  6. Using strategy rm

  7. Applied associate-*l/0.1

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1.0}}{x + y}}\]
  8. Using strategy rm

  9. Applied expm1-log1p-u0.1

    \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}{x + y}\]

  10. Final simplification0.1

    \[\leadsto \frac{\frac{y}{\left(y + x\right) + 1.0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y + x}\right)\right)}{y + x}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))