Average Error: 0.2 → 0.0
Time: 16.9s
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 r606386 = 6.0;
        double r606387 = x;
        double r606388 = 1.0;
        double r606389 = r606387 - r606388;
        double r606390 = r606386 * r606389;
        double r606391 = r606387 + r606388;
        double r606392 = 4.0;
        double r606393 = sqrt(r606387);
        double r606394 = r606392 * r606393;
        double r606395 = r606391 + r606394;
        double r606396 = r606390 / r606395;
        return r606396;
}
↓
double f(double x) {
        double r606397 = 6.0;
        double r606398 = x;
        double r606399 = 1.0;
        double r606400 = r606398 - r606399;
        double r606401 = sqrt(r606398);
        double r606402 = 4.0;
        double r606403 = r606398 + r606399;
        double r606404 = fma(r606401, r606402, r606403);
        double r606405 = r606400 / r606404;
        double r606406 = r606397 * r606405;
        return r606406;
}

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

    \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]
  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 2019323 +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)))))