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Target

Derivation

1. Initial program 30.0

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

3. Applied add-cube-cbrt30.0

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

4. Applied sqrt-prod30.0

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

5. Simplified30.0

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

7. Applied add-cube-cbrt30.0

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

8. Applied cbrt-prod29.9

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

10. Applied add-sqr-sqrt29.9

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

11. Final simplification29.9

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

Runtime

Time bar (total: 17.1s)Debug logProfile

herbie shell --seed 2018349 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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