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Target

Derivation

1. Initial program 29.9

   \[\sqrt{x + 1} - \sqrt{x}\]

2. Initial simplification29.9

   \[\leadsto \sqrt{1 + x} - \sqrt{x}\]
3. Using strategy rm

4. Applied add-cube-cbrt29.9

   \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \sqrt[3]{\sqrt{1 + x}}} - \sqrt{x}\]
5. Using strategy rm

6. Applied add-sqr-sqrt29.9

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

7. Applied cbrt-prod29.8

   \[\leadsto \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right)} - \sqrt{x}\]
8. Using strategy rm

9. Applied add-cube-cbrt29.8

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

10. Applied sqrt-prod29.8

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

11. Applied cbrt-prod29.8

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

12. Simplified29.8

    \[\leadsto \left(\sqrt[3]{\sqrt{1 + x}} \cdot \left(\color{blue}{\sqrt[3]{\left|\sqrt[3]{x + 1}\right|}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{1 + x}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right) - \sqrt{x}\]
13. Using strategy rm

14. Applied add-cube-cbrt29.8

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

15. Applied cbrt-prod29.8

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

16. Final simplification29.8

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

Runtime

Time bar (total: 21.0s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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