Average Error: 0.2 → 0.1
Time: 36.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}{6}} - \frac{1}{\frac{1}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}{6}} - \frac{1}{\frac{1}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}}}
double f(double x) {
        double r32502739 = 6.0;
        double r32502740 = x;
        double r32502741 = 1.0;
        double r32502742 = r32502740 - r32502741;
        double r32502743 = r32502739 * r32502742;
        double r32502744 = r32502740 + r32502741;
        double r32502745 = 4.0;
        double r32502746 = sqrt(r32502740);
        double r32502747 = r32502745 * r32502746;
        double r32502748 = r32502744 + r32502747;
        double r32502749 = r32502743 / r32502748;
        return r32502749;
}
↓
double f(double x) {
        double r32502750 = x;
        double r32502751 = sqrt(r32502750);
        double r32502752 = 4.0;
        double r32502753 = 1.0;
        double r32502754 = r32502753 + r32502750;
        double r32502755 = fma(r32502751, r32502752, r32502754);
        double r32502756 = 6.0;
        double r32502757 = r32502755 / r32502756;
        double r32502758 = r32502750 / r32502757;
        double r32502759 = 1.0;
        double r32502760 = r32502756 / r32502755;
        double r32502761 = r32502759 / r32502760;
        double r32502762 = r32502753 / r32502761;
        double r32502763 = r32502758 - r32502762;
        return r32502763;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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

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

  4. Applied div-sub0.1

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

  6. Applied clear-num0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))