Average Error: 0.2 → 0.1
Time: 9.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
double f(double x) {
        double r893766 = 6.0;
        double r893767 = x;
        double r893768 = 1.0;
        double r893769 = r893767 - r893768;
        double r893770 = r893766 * r893769;
        double r893771 = r893767 + r893768;
        double r893772 = 4.0;
        double r893773 = sqrt(r893767);
        double r893774 = r893772 * r893773;
        double r893775 = r893771 + r893774;
        double r893776 = r893770 / r893775;
        return r893776;
}
↓
double f(double x) {
        double r893777 = 6.0;
        double r893778 = x;
        double r893779 = 1.0;
        double r893780 = r893778 - r893779;
        double r893781 = r893778 + r893779;
        double r893782 = 4.0;
        double r893783 = sqrt(r893778);
        double r893784 = r893782 * r893783;
        double r893785 = r893781 + r893784;
        double r893786 = r893780 / r893785;
        double r893787 = r893777 * r893786;
        return r893787;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied add-cbrt-cube21.0

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

  4. Applied add-cbrt-cube21.7

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

  5. Applied add-cbrt-cube22.2

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

  6. Applied cbrt-unprod22.2

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

  7. Applied cbrt-undiv22.2

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

  8. Simplified1.0

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

  10. Applied *-un-lft-identity1.0

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

  11. Applied times-frac1.0

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

  12. Applied unpow-prod-down1.0

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

  13. Applied cbrt-prod1.0

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

  14. Simplified0.1

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 
(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)))))