Average Error: 0.2 → 0.1
Time: 6.8s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} \cdot \sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} \cdot \sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}
double f(double x) {
        double r1474688 = 6.0;
        double r1474689 = x;
        double r1474690 = 1.0;
        double r1474691 = r1474689 - r1474690;
        double r1474692 = r1474688 * r1474691;
        double r1474693 = r1474689 + r1474690;
        double r1474694 = 4.0;
        double r1474695 = sqrt(r1474689);
        double r1474696 = r1474694 * r1474695;
        double r1474697 = r1474693 + r1474696;
        double r1474698 = r1474692 / r1474697;
        return r1474698;
}
↓
double f(double x) {
        double r1474699 = x;
        double r1474700 = sqrt(r1474699);
        double r1474701 = 4.0;
        double r1474702 = 1.0;
        double r1474703 = r1474699 + r1474702;
        double r1474704 = fma(r1474700, r1474701, r1474703);
        double r1474705 = 6.0;
        double r1474706 = r1474704 / r1474705;
        double r1474707 = r1474699 / r1474706;
        double r1474708 = sqrt(r1474706);
        double r1474709 = r1474708 * r1474708;
        double r1474710 = r1474702 / r1474709;
        double r1474711 = r1474707 - r1474710;
        return r1474711;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

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-sub0.0

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

  6. Applied add-sqr-sqrt0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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)))))