Average Error: 19.8 → 0.8
Time: 4.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{{\left(-\sqrt{x + 1}\right)}^{3} - {\left(\sqrt{x}\right)}^{3}} \cdot \left(\left(-\sqrt{x + 1}\right) \cdot \left(-\sqrt{x + 1}\right) + \left(\sqrt{x} \cdot \sqrt{x} + \left(-\sqrt{x + 1}\right) \cdot \sqrt{x}\right)\right)}{\left(-\sqrt{x}\right) \cdot \frac{\sqrt{x + 1}}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{{\left(-\sqrt{x + 1}\right)}^{3} - {\left(\sqrt{x}\right)}^{3}} \cdot \left(\left(-\sqrt{x + 1}\right) \cdot \left(-\sqrt{x + 1}\right) + \left(\sqrt{x} \cdot \sqrt{x} + \left(-\sqrt{x + 1}\right) \cdot \sqrt{x}\right)\right)}{\left(-\sqrt{x}\right) \cdot \frac{\sqrt{x + 1}}{1}}
double code(double x) {
	return ((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0))));
}
↓
double code(double x) {
	return (((1.0 / (pow(-sqrt((x + 1.0)), 3.0) - pow(sqrt(x), 3.0))) * ((-sqrt((x + 1.0)) * -sqrt((x + 1.0))) + ((sqrt(x) * sqrt(x)) + (-sqrt((x + 1.0)) * sqrt(x))))) / (-sqrt(x) * (sqrt((x + 1.0)) / 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.8
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied clear-num19.8

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

  4. Applied frac-2neg19.8

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

  5. Applied frac-sub19.8

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

  6. Simplified19.8

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

  8. Applied flip-+19.6

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

  9. Simplified0.4

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

  11. Applied flip3--0.8

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

  12. Applied associate-/r/0.8

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

  13. Simplified0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2020071 
(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)))))