Average Error: 0.2 → 0.1
Time: 3.4s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{1}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{1}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{x - 1}}
(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (* 6.0 (/ 1.0 (/ (+ x (+ 1.0 (* 4.0 (sqrt x)))) (- x 1.0)))))
double code(double x) {
	return (((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 * (1.0 / (((double) (x + ((double) (1.0 + ((double) (4.0 * ((double) sqrt(x)))))))) / ((double) (x - 1.0))))));
}

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 Error: 0.2 bits

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

  2. SimplifiedError: 0.0 bits

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

  4. Applied clear-numError: 0.1 bits

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

  5. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))