Average Error: 62.4 → 57.8
Time: 21.7s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{4} + \log \left(\sqrt{\sqrt{e^{\frac{1}{\sqrt{x}}}}}\right)\right) + \log \left(\sqrt{e^{\frac{1}{\sqrt{x}}}}\right)\right) - \frac{1}{\sqrt{x + 1}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original62.4
Target62.6
Herbie57.8
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 62.4

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

  3. Applied add-log-exp58.2

    \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt58.2

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

  6. Applied log-prod58.2

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

  8. Applied add-sqr-sqrt58.2

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

  9. Applied sqrt-prod58.1

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

  10. Applied log-prod58.2

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

  11. Taylor expanded around 0 58.2

    \[\leadsto \left(\left(\log \left(\sqrt{\sqrt{e^{\frac{1}{\sqrt{x}}}}}\right) + \color{blue}{\log \left({\left(e^{\sqrt{\frac{1}{x}}}\right)}^{\frac{1}{4}}\right)}\right) + \log \left(\sqrt{e^{\frac{1}{\sqrt{x}}}}\right)\right) - \frac{1}{\sqrt{x + 1}}\]

  12. Simplified57.8

    \[\leadsto \left(\left(\log \left(\sqrt{\sqrt{e^{\frac{1}{\sqrt{x}}}}}\right) + \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{4}}\right) + \log \left(\sqrt{e^{\frac{1}{\sqrt{x}}}}\right)\right) - \frac{1}{\sqrt{x + 1}}\]

  13. Final simplification57.8

    \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{4} + \log \left(\sqrt{\sqrt{e^{\frac{1}{\sqrt{x}}}}}\right)\right) + \log \left(\sqrt{e^{\frac{1}{\sqrt{x}}}}\right)\right) - \frac{1}{\sqrt{x + 1}}\]

Runtime

Time bar (total: 21.7s)Debug logProfile

herbie shell --seed 2018255 
(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)))))