Average Error: 0.2 → 0.1
Time: 6.6s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot \frac{6}{\frac{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{\sqrt{x} - \sqrt{1}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot \frac{6}{\frac{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{\sqrt{x} - \sqrt{1}}}
double f(double x) {
        double r811478 = 6.0;
        double r811479 = x;
        double r811480 = 1.0;
        double r811481 = r811479 - r811480;
        double r811482 = r811478 * r811481;
        double r811483 = r811479 + r811480;
        double r811484 = 4.0;
        double r811485 = sqrt(r811479);
        double r811486 = r811484 * r811485;
        double r811487 = r811483 + r811486;
        double r811488 = r811482 / r811487;
        return r811488;
}
↓
double f(double x) {
        double r811489 = x;
        double r811490 = sqrt(r811489);
        double r811491 = 1.0;
        double r811492 = sqrt(r811491);
        double r811493 = r811490 + r811492;
        double r811494 = r811489 + r811491;
        double r811495 = 4.0;
        double r811496 = r811495 * r811490;
        double r811497 = r811494 + r811496;
        double r811498 = sqrt(r811497);
        double r811499 = r811493 / r811498;
        double r811500 = 6.0;
        double r811501 = r811490 - r811492;
        double r811502 = r811498 / r811501;
        double r811503 = r811500 / r811502;
        double r811504 = r811499 * r811503;
        return r811504;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.2
Target0.0
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. Using strategy rm

  3. Applied associate-/l*0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied difference-of-squares0.3

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied times-frac0.1

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

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

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

  11. Applied times-frac0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 
(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)))))