Average Error: 0.2 → 0.1
Time: 6.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r926550 = 6.0;
        double r926551 = x;
        double r926552 = 1.0;
        double r926553 = r926551 - r926552;
        double r926554 = r926550 * r926553;
        double r926555 = r926551 + r926552;
        double r926556 = 4.0;
        double r926557 = sqrt(r926551);
        double r926558 = r926556 * r926557;
        double r926559 = r926555 + r926558;
        double r926560 = r926554 / r926559;
        return r926560;
}
↓
double f(double x) {
        double r926561 = x;
        double r926562 = sqrt(r926561);
        double r926563 = 4.0;
        double r926564 = 1.0;
        double r926565 = r926561 + r926564;
        double r926566 = fma(r926562, r926563, r926565);
        double r926567 = 6.0;
        double r926568 = r926566 / r926567;
        double r926569 = r926561 / r926568;
        double r926570 = r926564 / r926566;
        double r926571 = 1.0;
        double r926572 = r926571 / r926567;
        double r926573 = r926570 / r926572;
        double r926574 = r926569 - r926573;
        return r926574;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

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 div-sub0.1

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

  6. Applied div-inv0.1

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 +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)))))