Average Error: 0.3 → 0.0
Time: 4.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r952300 = 6.0;
        double r952301 = x;
        double r952302 = 1.0;
        double r952303 = r952301 - r952302;
        double r952304 = r952300 * r952303;
        double r952305 = r952301 + r952302;
        double r952306 = 4.0;
        double r952307 = sqrt(r952301);
        double r952308 = r952306 * r952307;
        double r952309 = r952305 + r952308;
        double r952310 = r952304 / r952309;
        return r952310;
}
↓
double f(double x) {
        double r952311 = x;
        double r952312 = 1.0;
        double r952313 = r952311 - r952312;
        double r952314 = sqrt(r952311);
        double r952315 = 4.0;
        double r952316 = r952311 + r952312;
        double r952317 = fma(r952314, r952315, r952316);
        double r952318 = r952313 / r952317;
        double r952319 = 1.0;
        double r952320 = 6.0;
        double r952321 = r952319 / r952320;
        double r952322 = r952318 / r952321;
        return r952322;
}

Error

Bits error versus x

Target

Original0.3
Target0.0
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.3

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

  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-inv0.2

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 +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)))))