Average Error: 0.2 → 0.0
Time: 5.1s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r834742 = 6.0;
        double r834743 = x;
        double r834744 = 1.0;
        double r834745 = r834743 - r834744;
        double r834746 = r834742 * r834745;
        double r834747 = r834743 + r834744;
        double r834748 = 4.0;
        double r834749 = sqrt(r834743);
        double r834750 = r834748 * r834749;
        double r834751 = r834747 + r834750;
        double r834752 = r834746 / r834751;
        return r834752;
}
↓
double f(double x) {
        double r834753 = x;
        double r834754 = 1.0;
        double r834755 = r834753 - r834754;
        double r834756 = sqrt(r834753);
        double r834757 = 4.0;
        double r834758 = r834753 + r834754;
        double r834759 = fma(r834756, r834757, r834758);
        double r834760 = 6.0;
        double r834761 = r834759 / r834760;
        double r834762 = r834755 / r834761;
        return r834762;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +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)))))