Average Error: 0.2 → 0.0
Time: 13.6s
Precision: 64
\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
↓
\[6.0 \cdot \frac{x - 1.0}{\mathsf{fma}\left(4.0, \sqrt{x}, x + 1.0\right)}\]
\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(4.0, \sqrt{x}, x + 1.0\right)}
double f(double x) {
        double r45164462 = 6.0;
        double r45164463 = x;
        double r45164464 = 1.0;
        double r45164465 = r45164463 - r45164464;
        double r45164466 = r45164462 * r45164465;
        double r45164467 = r45164463 + r45164464;
        double r45164468 = 4.0;
        double r45164469 = sqrt(r45164463);
        double r45164470 = r45164468 * r45164469;
        double r45164471 = r45164467 + r45164470;
        double r45164472 = r45164466 / r45164471;
        return r45164472;
}
↓
double f(double x) {
        double r45164473 = 6.0;
        double r45164474 = x;
        double r45164475 = 1.0;
        double r45164476 = r45164474 - r45164475;
        double r45164477 = 4.0;
        double r45164478 = sqrt(r45164474);
        double r45164479 = r45164474 + r45164475;
        double r45164480 = fma(r45164477, r45164478, r45164479);
        double r45164481 = r45164476 / r45164480;
        double r45164482 = r45164473 * r45164481;
        return r45164482;
}

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 div-inv0.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +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)))))