Average Error: 0.2 → 0.1
Time: 10.7s
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 r910207 = 6.0;
        double r910208 = x;
        double r910209 = 1.0;
        double r910210 = r910208 - r910209;
        double r910211 = r910207 * r910210;
        double r910212 = r910208 + r910209;
        double r910213 = 4.0;
        double r910214 = sqrt(r910208);
        double r910215 = r910213 * r910214;
        double r910216 = r910212 + r910215;
        double r910217 = r910211 / r910216;
        return r910217;
}
↓
double f(double x) {
        double r910218 = x;
        double r910219 = sqrt(r910218);
        double r910220 = 1.0;
        double r910221 = sqrt(r910220);
        double r910222 = r910219 + r910221;
        double r910223 = r910218 + r910220;
        double r910224 = 4.0;
        double r910225 = r910224 * r910219;
        double r910226 = r910223 + r910225;
        double r910227 = sqrt(r910226);
        double r910228 = r910222 / r910227;
        double r910229 = 6.0;
        double r910230 = r910219 - r910221;
        double r910231 = r910227 / r910230;
        double r910232 = r910229 / r910231;
        double r910233 = r910228 * r910232;
        return r910233;
}

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)))))