Average Error: 19.6 → 0.4
Time: 5.6s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(\frac{1}{x} \cdot 1\right) \cdot \frac{1}{\left(\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{x + 1}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(\frac{1}{x} \cdot 1\right) \cdot \frac{1}{\left(\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{x + 1}{1}}
double code(double x) {
	return ((double) ((1.0 / ((double) sqrt(x))) - (1.0 / ((double) sqrt(((double) (x + 1.0)))))));
}
↓
double code(double x) {
	return ((double) (((double) ((1.0 / x) * 1.0)) * (1.0 / ((double) (((double) ((1.0 / ((double) sqrt(((double) (x + 1.0))))) + (1.0 / ((double) sqrt(x))))) * (((double) (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

Original19.6
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied flip--19.6

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

  4. Simplified19.7

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

  5. Simplified19.7

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

  7. Applied frac-sub19.1

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

  8. Applied associate-/l/19.1

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

  9. Simplified19.0

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

  10. Taylor expanded around 0 0.7

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

  12. Applied *-un-lft-identity0.7

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

  13. Applied times-frac0.4

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

  14. Simplified0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020199 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))