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Target

Derivation

1. Initial program 15.2

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

2. Simplified15.2

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

4. Applied *-un-lft-identity15.2

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

5. Applied associate-/r*15.2

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

7. Applied add-sqr-sqrt15.2

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

8. Applied div-inv15.2

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

9. Applied times-frac15.1

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

10. Simplified15.1

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

11. Simplified0.0

    \[\leadsto \frac{x}{\sqrt{1^2 + x^2}^*} \cdot \color{blue}{\frac{1}{\sqrt{1^2 + x^2}^*}}\]

12. Final simplification0.0

    \[\leadsto \frac{x}{\sqrt{1^2 + x^2}^*} \cdot \frac{1}{\sqrt{1^2 + x^2}^*}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 6.5s)Debug log