Average Error: 0.2 → 0.1
Time: 5.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \left(\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}}\right)\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \left(\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\sqrt{\left(x + 1\right) + 4 \cdot \sqrt{x}}}\right)
double code(double x) {
	return ((double) (((double) (6.0 * ((double) (x - 1.0)))) / ((double) (((double) (x + 1.0)) + ((double) (4.0 * ((double) sqrt(x))))))));
}
↓
double code(double x) {
	return ((double) (6.0 * ((double) (((double) (((double) (((double) sqrt(x)) + ((double) sqrt(1.0)))) / ((double) sqrt(((double) (((double) (x + 1.0)) + ((double) (4.0 * ((double) sqrt(x)))))))))) * ((double) (((double) (((double) sqrt(x)) - ((double) sqrt(1.0)))) / ((double) sqrt(((double) (((double) (x + 1.0)) + ((double) (4.0 * ((double) sqrt(x))))))))))))));
}

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 *-un-lft-identity0.2

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

  4. Applied times-frac0.0

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

  5. Simplified0.0

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied add-sqr-sqrt0.1

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

  10. Applied difference-of-squares0.1

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

  11. Applied times-frac0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020123 
(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)))))