Average Error: 0.2 → 0.0
Time: 10.1s
Precision: 64
\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
↓
\[6.0 \cdot \frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\]
\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}
↓
6.0 \cdot \frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}
double f(double x) {
        double r16078631 = 6.0;
        double r16078632 = x;
        double r16078633 = 1.0;
        double r16078634 = r16078632 - r16078633;
        double r16078635 = r16078631 * r16078634;
        double r16078636 = r16078632 + r16078633;
        double r16078637 = 4.0;
        double r16078638 = sqrt(r16078632);
        double r16078639 = r16078637 * r16078638;
        double r16078640 = r16078636 + r16078639;
        double r16078641 = r16078635 / r16078640;
        return r16078641;
}
↓
double f(double x) {
        double r16078642 = 6.0;
        double r16078643 = x;
        double r16078644 = 1.0;
        double r16078645 = r16078643 - r16078644;
        double r16078646 = sqrt(r16078643);
        double r16078647 = 4.0;
        double r16078648 = r16078643 + r16078644;
        double r16078649 = fma(r16078646, r16078647, r16078648);
        double r16078650 = r16078645 / r16078649;
        double r16078651 = r16078642 * r16078650;
        return r16078651;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6.0}{\frac{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}{x - 1.0}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{6.0}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}}\]
  3. Using strategy rm

  4. Applied clear-num0.0

    \[\leadsto \frac{6.0}{\color{blue}{\frac{1}{\frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}}}}\]
  5. Using strategy rm

  6. Applied associate-/r/0.0

    \[\leadsto \color{blue}{\frac{6.0}{1} \cdot \frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}}\]

  7. Simplified0.0

    \[\leadsto \color{blue}{6.0} \cdot \frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\]

  8. Final simplification0.0

    \[\leadsto 6.0 \cdot \frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\]

Reproduce

herbie shell --seed 2019156 +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)))))