Average Error: 0.2 → 0.0
Time: 35.8s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}
double f(double x) {
        double r629921 = 6.0;
        double r629922 = x;
        double r629923 = 1.0;
        double r629924 = r629922 - r629923;
        double r629925 = r629921 * r629924;
        double r629926 = r629922 + r629923;
        double r629927 = 4.0;
        double r629928 = sqrt(r629922);
        double r629929 = r629927 * r629928;
        double r629930 = r629926 + r629929;
        double r629931 = r629925 / r629930;
        return r629931;
}
↓
double f(double x) {
        double r629932 = 6.0;
        double r629933 = x;
        double r629934 = 1.0;
        double r629935 = r629933 - r629934;
        double r629936 = 4.0;
        double r629937 = sqrt(r629933);
        double r629938 = r629933 + r629934;
        double r629939 = fma(r629936, r629937, r629938);
        double r629940 = r629935 / r629939;
        double r629941 = r629932 * r629940;
        return r629941;
}

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-inv0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +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)))))