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 r1016968 = 6.0;
        double r1016969 = x;
        double r1016970 = 1.0;
        double r1016971 = r1016969 - r1016970;
        double r1016972 = r1016968 * r1016971;
        double r1016973 = r1016969 + r1016970;
        double r1016974 = 4.0;
        double r1016975 = sqrt(r1016969);
        double r1016976 = r1016974 * r1016975;
        double r1016977 = r1016973 + r1016976;
        double r1016978 = r1016972 / r1016977;
        return r1016978;
}
↓
double f(double x) {
        double r1016979 = x;
        double r1016980 = sqrt(r1016979);
        double r1016981 = 1.0;
        double r1016982 = sqrt(r1016981);
        double r1016983 = r1016980 + r1016982;
        double r1016984 = r1016979 + r1016981;
        double r1016985 = 4.0;
        double r1016986 = r1016985 * r1016980;
        double r1016987 = r1016984 + r1016986;
        double r1016988 = sqrt(r1016987);
        double r1016989 = r1016983 / r1016988;
        double r1016990 = 6.0;
        double r1016991 = r1016980 - r1016982;
        double r1016992 = r1016988 / r1016991;
        double r1016993 = r1016990 / r1016992;
        double r1016994 = r1016989 * r1016993;
        return r1016994;
}

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)))))