Average Error: 19.8 → 0.6
Time: 18.3s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{1}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{x + 1} + \sqrt{x}}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Applied associate-/l/19.6

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

  8. Simplified0.7

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

  10. Applied associate-/r*0.4

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

  12. Applied add-cube-cbrt0.6

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

  13. Applied *-un-lft-identity0.6

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

  14. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}}}}\]

  15. Final simplification0.6

    \[\leadsto \frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{1}{\sqrt[3]{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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