Average Error: 0.2 → 0.0
Time: 3.5s
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 r760616 = 6.0;
        double r760617 = x;
        double r760618 = 1.0;
        double r760619 = r760617 - r760618;
        double r760620 = r760616 * r760619;
        double r760621 = r760617 + r760618;
        double r760622 = 4.0;
        double r760623 = sqrt(r760617);
        double r760624 = r760622 * r760623;
        double r760625 = r760621 + r760624;
        double r760626 = r760620 / r760625;
        return r760626;
}
↓
double f(double x) {
        double r760627 = x;
        double r760628 = 1.0;
        double r760629 = r760627 - r760628;
        double r760630 = sqrt(r760627);
        double r760631 = 4.0;
        double r760632 = r760627 + r760628;
        double r760633 = fma(r760630, r760631, r760632);
        double r760634 = r760629 / r760633;
        double r760635 = 6.0;
        double r760636 = r760634 * r760635;
        return r760636;
}

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)))))