Average Error: 29.4 → 0.3
Time: 4.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}
double code(double x) {
	return (sqrt((x + 1.0)) - sqrt(x));
}
↓
double code(double x) {
	return (sqrt((1.0 / (sqrt((x + 1.0)) + sqrt(x)))) * sqrt((1.0 / (sqrt((x + 1.0)) + sqrt(x)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.2

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}\]

  7. Final simplification0.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))