Average Error: 0.2 → 0.0
Time: 3.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} - \frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right) \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} - \frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right) \cdot 6
double f(double x) {
        double r740433 = 6.0;
        double r740434 = x;
        double r740435 = 1.0;
        double r740436 = r740434 - r740435;
        double r740437 = r740433 * r740436;
        double r740438 = r740434 + r740435;
        double r740439 = 4.0;
        double r740440 = sqrt(r740434);
        double r740441 = r740439 * r740440;
        double r740442 = r740438 + r740441;
        double r740443 = r740437 / r740442;
        return r740443;
}
↓
double f(double x) {
        double r740444 = x;
        double r740445 = sqrt(r740444);
        double r740446 = 4.0;
        double r740447 = 1.0;
        double r740448 = r740444 + r740447;
        double r740449 = fma(r740445, r740446, r740448);
        double r740450 = r740444 / r740449;
        double r740451 = r740447 / r740449;
        double r740452 = r740450 - r740451;
        double r740453 = 6.0;
        double r740454 = r740452 * r740453;
        return r740454;
}

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. 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. Using strategy rm

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 +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)))))