Average Error: 0.2 → 0.0
Time: 3.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6
double f(double x) {
        double r861852 = 6.0;
        double r861853 = x;
        double r861854 = 1.0;
        double r861855 = r861853 - r861854;
        double r861856 = r861852 * r861855;
        double r861857 = r861853 + r861854;
        double r861858 = 4.0;
        double r861859 = sqrt(r861853);
        double r861860 = r861858 * r861859;
        double r861861 = r861857 + r861860;
        double r861862 = r861856 / r861861;
        return r861862;
}
↓
double f(double x) {
        double r861863 = x;
        double r861864 = 1.0;
        double r861865 = r861863 - r861864;
        double r861866 = sqrt(r861863);
        double r861867 = 4.0;
        double r861868 = r861863 + r861864;
        double r861869 = fma(r861866, r861867, r861868);
        double r861870 = r861865 / r861869;
        double r861871 = 6.0;
        double r861872 = r861870 * r861871;
        return r861872;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  3. Using strategy rm

  4. Applied associate-/r/0.0

    \[\leadsto \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6}\]

  5. Final simplification0.0

    \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))