Average Error: 19.6 → 0.8
Time: 47.2s
Precision: 64
Internal Precision: 1088
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \left(1 + x\right)}}\]

Error

Bits error versus x

Target

Original19.6
Target0.7
Herbie0.8
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Applied simplify19.6

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

  6. Applied flip--19.4

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

  7. Applied simplify0.4

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

  9. Applied add-cbrt-cube0.9

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

  10. Applied simplify0.8

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

Runtime

Time bar (total: 47.2s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +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)))))