Average Error: 0.2 → 0.0
Time: 33.8s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r36586479 = 6.0;
        double r36586480 = x;
        double r36586481 = 1.0;
        double r36586482 = r36586480 - r36586481;
        double r36586483 = r36586479 * r36586482;
        double r36586484 = r36586480 + r36586481;
        double r36586485 = 4.0;
        double r36586486 = sqrt(r36586480);
        double r36586487 = r36586485 * r36586486;
        double r36586488 = r36586484 + r36586487;
        double r36586489 = r36586483 / r36586488;
        return r36586489;
}
↓
double f(double x) {
        double r36586490 = 6.0;
        double r36586491 = x;
        double r36586492 = 1.0;
        double r36586493 = r36586491 - r36586492;
        double r36586494 = sqrt(r36586491);
        double r36586495 = 4.0;
        double r36586496 = r36586491 + r36586492;
        double r36586497 = fma(r36586494, r36586495, r36586496);
        double r36586498 = r36586493 / r36586497;
        double r36586499 = r36586490 * r36586498;
        return r36586499;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

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)))))