Average Error: 0.2 → 0.1
Time: 9.3s
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 r992975 = 6.0;
        double r992976 = x;
        double r992977 = 1.0;
        double r992978 = r992976 - r992977;
        double r992979 = r992975 * r992978;
        double r992980 = r992976 + r992977;
        double r992981 = 4.0;
        double r992982 = sqrt(r992976);
        double r992983 = r992981 * r992982;
        double r992984 = r992980 + r992983;
        double r992985 = r992979 / r992984;
        return r992985;
}
↓
double f(double x) {
        double r992986 = x;
        double r992987 = sqrt(r992986);
        double r992988 = 1.0;
        double r992989 = sqrt(r992988);
        double r992990 = r992987 + r992989;
        double r992991 = r992986 + r992988;
        double r992992 = 4.0;
        double r992993 = r992992 * r992987;
        double r992994 = r992991 + r992993;
        double r992995 = sqrt(r992994);
        double r992996 = r992990 / r992995;
        double r992997 = 6.0;
        double r992998 = r992987 - r992989;
        double r992999 = r992995 / r992998;
        double r993000 = r992997 / r992999;
        double r993001 = r992996 * r993000;
        return r993001;
}

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 associate-/l*0.1

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

  5. Applied add-sqr-sqrt0.1

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

  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 2020046 
(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)))))