Average Error: 0.3 → 0.0
Time: 13.3s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \cdot 6
double f(double x) {
        double r37940858 = 6.0;
        double r37940859 = x;
        double r37940860 = 1.0;
        double r37940861 = r37940859 - r37940860;
        double r37940862 = r37940858 * r37940861;
        double r37940863 = r37940859 + r37940860;
        double r37940864 = 4.0;
        double r37940865 = sqrt(r37940859);
        double r37940866 = r37940864 * r37940865;
        double r37940867 = r37940863 + r37940866;
        double r37940868 = r37940862 / r37940867;
        return r37940868;
}
↓
double f(double x) {
        double r37940869 = x;
        double r37940870 = 1.0;
        double r37940871 = r37940869 - r37940870;
        double r37940872 = 4.0;
        double r37940873 = sqrt(r37940869);
        double r37940874 = r37940870 + r37940869;
        double r37940875 = fma(r37940872, r37940873, r37940874);
        double r37940876 = r37940871 / r37940875;
        double r37940877 = 6.0;
        double r37940878 = r37940876 * r37940877;
        return r37940878;
}

Error

Bits error versus x

Target

Original0.3
Target0.1
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

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

  4. Applied clear-num0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied *-un-lft-identity0.1

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

  8. Applied times-frac0.1

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

  9. Applied add-cube-cbrt0.1

    \[\leadsto 6 \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}\]

  10. Applied times-frac0.1

    \[\leadsto 6 \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}\right)}\]

  11. Simplified0.1

    \[\leadsto 6 \cdot \left(\color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}\right)\]

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))