Average Error: 0.2 → 0.1
Time: 4.3s
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 r1135821 = 6.0;
        double r1135822 = x;
        double r1135823 = 1.0;
        double r1135824 = r1135822 - r1135823;
        double r1135825 = r1135821 * r1135824;
        double r1135826 = r1135822 + r1135823;
        double r1135827 = 4.0;
        double r1135828 = sqrt(r1135822);
        double r1135829 = r1135827 * r1135828;
        double r1135830 = r1135826 + r1135829;
        double r1135831 = r1135825 / r1135830;
        return r1135831;
}
↓
double f(double x) {
        double r1135832 = x;
        double r1135833 = 1.0;
        double r1135834 = r1135832 - r1135833;
        double r1135835 = sqrt(r1135832);
        double r1135836 = 4.0;
        double r1135837 = r1135832 + r1135833;
        double r1135838 = fma(r1135835, r1135836, r1135837);
        double r1135839 = r1135834 / r1135838;
        double r1135840 = 1.0;
        double r1135841 = 6.0;
        double r1135842 = r1135840 / r1135841;
        double r1135843 = r1135839 / r1135842;
        return r1135843;
}

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. Simplified0.1

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  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*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019362 +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)))))