Average Error: 0.2 → 0.0
Time: 14.8s
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 r1022689 = 6.0;
        double r1022690 = x;
        double r1022691 = 1.0;
        double r1022692 = r1022690 - r1022691;
        double r1022693 = r1022689 * r1022692;
        double r1022694 = r1022690 + r1022691;
        double r1022695 = 4.0;
        double r1022696 = sqrt(r1022690);
        double r1022697 = r1022695 * r1022696;
        double r1022698 = r1022694 + r1022697;
        double r1022699 = r1022693 / r1022698;
        return r1022699;
}
↓
double f(double x) {
        double r1022700 = x;
        double r1022701 = 1.0;
        double r1022702 = r1022700 - r1022701;
        double r1022703 = sqrt(r1022700);
        double r1022704 = 4.0;
        double r1022705 = r1022700 + r1022701;
        double r1022706 = fma(r1022703, r1022704, r1022705);
        double r1022707 = r1022702 / r1022706;
        double r1022708 = 6.0;
        double r1022709 = r1022707 * r1022708;
        return r1022709;
}

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)))))