Average Error: 19.9 → 0.5
Time: 5.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \sqrt[3]{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{3}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \sqrt[3]{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{3}}}{\sqrt{x} \cdot \sqrt{x + 1}}
double code(double x) {
	return ((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0))));
}
↓
double code(double x) {
	return ((1.0 * cbrt(pow((1.0 / (sqrt((x + 1.0)) + sqrt(x))), 3.0))) / (sqrt(x) * sqrt((x + 1.0))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied frac-sub19.9

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

  4. Simplified19.9

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

  6. Applied flip--19.6

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

  7. Simplified19.2

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-cbrt-cube0.9

    \[\leadsto \frac{1 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  11. Applied add-cbrt-cube0.9

    \[\leadsto \frac{1 \cdot \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  12. Applied cbrt-undiv0.9

    \[\leadsto \frac{1 \cdot \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  13. Simplified0.5

    \[\leadsto \frac{1 \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{3}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  14. Final simplification0.5

    \[\leadsto \frac{1 \cdot \sqrt[3]{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{3}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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