Average Error: 0.2 → 0.0
Time: 10.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r36169872 = 6.0;
        double r36169873 = x;
        double r36169874 = 1.0;
        double r36169875 = r36169873 - r36169874;
        double r36169876 = r36169872 * r36169875;
        double r36169877 = r36169873 + r36169874;
        double r36169878 = 4.0;
        double r36169879 = sqrt(r36169873);
        double r36169880 = r36169878 * r36169879;
        double r36169881 = r36169877 + r36169880;
        double r36169882 = r36169876 / r36169881;
        return r36169882;
}
↓
double f(double x) {
        double r36169883 = 6.0;
        double r36169884 = x;
        double r36169885 = 1.0;
        double r36169886 = r36169884 - r36169885;
        double r36169887 = sqrt(r36169884);
        double r36169888 = 4.0;
        double r36169889 = r36169884 + r36169885;
        double r36169890 = fma(r36169887, r36169888, r36169889);
        double r36169891 = r36169886 / r36169890;
        double r36169892 = r36169883 * r36169891;
        return r36169892;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))