Average Error: 0.2 → 0.0
Time: 1.3m
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r1120689 = 6.0;
        double r1120690 = x;
        double r1120691 = 1.0;
        double r1120692 = r1120690 - r1120691;
        double r1120693 = r1120689 * r1120692;
        double r1120694 = r1120690 + r1120691;
        double r1120695 = 4.0;
        double r1120696 = sqrt(r1120690);
        double r1120697 = r1120695 * r1120696;
        double r1120698 = r1120694 + r1120697;
        double r1120699 = r1120693 / r1120698;
        return r1120699;
}
↓
double f(double x) {
        double r1120700 = x;
        double r1120701 = 1.0;
        double r1120702 = r1120700 - r1120701;
        double r1120703 = sqrt(r1120700);
        double r1120704 = 4.0;
        double r1120705 = r1120700 + r1120701;
        double r1120706 = fma(r1120703, r1120704, r1120705);
        double r1120707 = r1120702 / r1120706;
        double r1120708 = 1.0;
        double r1120709 = 6.0;
        double r1120710 = r1120708 / r1120709;
        double r1120711 = r1120707 / r1120710;
        return r1120711;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
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 div-inv0.2

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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)))))