Average Error: 0.2 → 0.1
Time: 1.2m
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{6}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{1} + \sqrt{x}}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} - \sqrt{1}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{6}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{1} + \sqrt{x}}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} - \sqrt{1}}}
double f(double x) {
        double r28444997 = 6.0;
        double r28444998 = x;
        double r28444999 = 1.0;
        double r28445000 = r28444998 - r28444999;
        double r28445001 = r28444997 * r28445000;
        double r28445002 = r28444998 + r28444999;
        double r28445003 = 4.0;
        double r28445004 = sqrt(r28444998);
        double r28445005 = r28445003 * r28445004;
        double r28445006 = r28445002 + r28445005;
        double r28445007 = r28445001 / r28445006;
        return r28445007;
}
↓
double f(double x) {
        double r28445008 = 6.0;
        double r28445009 = x;
        double r28445010 = sqrt(r28445009);
        double r28445011 = 4.0;
        double r28445012 = 1.0;
        double r28445013 = r28445009 + r28445012;
        double r28445014 = fma(r28445010, r28445011, r28445013);
        double r28445015 = sqrt(r28445014);
        double r28445016 = sqrt(r28445012);
        double r28445017 = r28445016 + r28445010;
        double r28445018 = r28445015 / r28445017;
        double r28445019 = r28445008 / r28445018;
        double r28445020 = r28445010 - r28445016;
        double r28445021 = r28445015 / r28445020;
        double r28445022 = r28445019 / r28445021;
        return r28445022;
}

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

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied difference-of-squares0.3

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied times-frac0.1

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +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)))))