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Target

Derivation

1. Initial program 14.6

   \[\frac{x}{x \cdot x + 1}\]

2. Initial simplification14.6

   \[\leadsto \frac{x}{(x \cdot x + 1)_*}\]
3. Using strategy rm

4. Applied add-sqr-sqrt14.6

   \[\leadsto \frac{x}{\color{blue}{\sqrt{(x \cdot x + 1)_*} \cdot \sqrt{(x \cdot x + 1)_*}}}\]

5. Applied associate-/r*14.5

   \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity14.5

   \[\leadsto \frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\color{blue}{1 \cdot \sqrt{(x \cdot x + 1)_*}}}\]

8. Applied add-cube-cbrt15.3

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

9. Applied times-frac15.3

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

10. Simplified15.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}} \cdot \sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}}\right)} \cdot \frac{\sqrt[3]{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}}{\sqrt{(x \cdot x + 1)_*}}\]

11. Simplified0.8

    \[\leadsto \left(\sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}} \cdot \sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}}}{\sqrt{1^2 + x^2}^*}}\]

12. Final simplification0.8

    \[\leadsto \left(\sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}} \cdot \sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}}\right) \cdot \frac{\sqrt[3]{\frac{x}{\sqrt{1^2 + x^2}^*}}}{\sqrt{1^2 + x^2}^*}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

  (/ x (+ (* x x) 1)))

Details

Time bar (total: 12.9s)Debug log