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 r541671 = 6.0;
        double r541672 = x;
        double r541673 = 1.0;
        double r541674 = r541672 - r541673;
        double r541675 = r541671 * r541674;
        double r541676 = r541672 + r541673;
        double r541677 = 4.0;
        double r541678 = sqrt(r541672);
        double r541679 = r541677 * r541678;
        double r541680 = r541676 + r541679;
        double r541681 = r541675 / r541680;
        return r541681;
}
↓
double f(double x) {
        double r541682 = 6.0;
        double r541683 = x;
        double r541684 = 1.0;
        double r541685 = r541683 - r541684;
        double r541686 = r541683 + r541684;
        double r541687 = 4.0;
        double r541688 = sqrt(r541683);
        double r541689 = r541687 * r541688;
        double r541690 = r541686 + r541689;
        double r541691 = r541685 / r541690;
        double r541692 = r541682 * r541691;
        return r541692;
}

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)))))