Average Error: 0.2 → 0.0
Time: 17.5s
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 r1153680 = 6.0;
        double r1153681 = x;
        double r1153682 = 1.0;
        double r1153683 = r1153681 - r1153682;
        double r1153684 = r1153680 * r1153683;
        double r1153685 = r1153681 + r1153682;
        double r1153686 = 4.0;
        double r1153687 = sqrt(r1153681);
        double r1153688 = r1153686 * r1153687;
        double r1153689 = r1153685 + r1153688;
        double r1153690 = r1153684 / r1153689;
        return r1153690;
}
↓
double f(double x) {
        double r1153691 = x;
        double r1153692 = 1.0;
        double r1153693 = r1153691 - r1153692;
        double r1153694 = sqrt(r1153691);
        double r1153695 = 4.0;
        double r1153696 = r1153691 + r1153692;
        double r1153697 = fma(r1153694, r1153695, r1153696);
        double r1153698 = r1153693 / r1153697;
        double r1153699 = 6.0;
        double r1153700 = r1153698 * r1153699;
        return r1153700;
}

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 2020060 +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)))))