Average Error: 19.3 → 0.1
Time: 19.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{\left(y + x\right) + 1.0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y + x}\right)\right)}{y + x}\]
\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 r20212808 = x;
        double r20212809 = y;
        double r20212810 = r20212808 * r20212809;
        double r20212811 = r20212808 + r20212809;
        double r20212812 = r20212811 * r20212811;
        double r20212813 = 1.0;
        double r20212814 = r20212811 + r20212813;
        double r20212815 = r20212812 * r20212814;
        double r20212816 = r20212810 / r20212815;
        return r20212816;
}
↓
double f(double x, double y) {
        double r20212817 = y;
        double r20212818 = x;
        double r20212819 = r20212817 + r20212818;
        double r20212820 = 1.0;
        double r20212821 = r20212819 + r20212820;
        double r20212822 = r20212817 / r20212821;
        double r20212823 = r20212818 / r20212819;
        double r20212824 = log1p(r20212823);
        double r20212825 = expm1(r20212824);
        double r20212826 = r20212822 * r20212825;
        double r20212827 = r20212826 / r20212819;
        return r20212827;
}

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))))