Average Error: 0.2 → 0.0
Time: 17.3s
Precision: 64
\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1.0}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, 1.0\right) + x}{6.0}}\]
\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}
↓
\frac{x - 1.0}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, 1.0\right) + x}{6.0}}
double f(double x) {
        double r37030346 = 6.0;
        double r37030347 = x;
        double r37030348 = 1.0;
        double r37030349 = r37030347 - r37030348;
        double r37030350 = r37030346 * r37030349;
        double r37030351 = r37030347 + r37030348;
        double r37030352 = 4.0;
        double r37030353 = sqrt(r37030347);
        double r37030354 = r37030352 * r37030353;
        double r37030355 = r37030351 + r37030354;
        double r37030356 = r37030350 / r37030355;
        return r37030356;
}
↓
double f(double x) {
        double r37030357 = x;
        double r37030358 = 1.0;
        double r37030359 = r37030357 - r37030358;
        double r37030360 = sqrt(r37030357);
        double r37030361 = 4.0;
        double r37030362 = fma(r37030360, r37030361, r37030358);
        double r37030363 = r37030362 + r37030357;
        double r37030364 = 6.0;
        double r37030365 = r37030363 / r37030364;
        double r37030366 = r37030359 / r37030365;
        return r37030366;
}

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. Simplified0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied associate-/r*0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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)))))