Average Error: 0.2 → 0.1
Time: 1.3m
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 r42114506 = 6.0;
        double r42114507 = x;
        double r42114508 = 1.0;
        double r42114509 = r42114507 - r42114508;
        double r42114510 = r42114506 * r42114509;
        double r42114511 = r42114507 + r42114508;
        double r42114512 = 4.0;
        double r42114513 = sqrt(r42114507);
        double r42114514 = r42114512 * r42114513;
        double r42114515 = r42114511 + r42114514;
        double r42114516 = r42114510 / r42114515;
        return r42114516;
}
↓
double f(double x) {
        double r42114517 = 6.0;
        double r42114518 = x;
        double r42114519 = sqrt(r42114518);
        double r42114520 = 4.0;
        double r42114521 = 1.0;
        double r42114522 = r42114518 + r42114521;
        double r42114523 = fma(r42114519, r42114520, r42114522);
        double r42114524 = sqrt(r42114523);
        double r42114525 = sqrt(r42114521);
        double r42114526 = r42114525 + r42114519;
        double r42114527 = r42114524 / r42114526;
        double r42114528 = r42114517 / r42114527;
        double r42114529 = r42114519 - r42114525;
        double r42114530 = r42114524 / r42114529;
        double r42114531 = r42114528 / r42114530;
        return r42114531;
}

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)))))