Average Error: 14.5 → 0.0
Time: 6.2s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{\sqrt{1^2 + x^2}^*} \cdot \frac{x}{\sqrt{1^2 + x^2}^*}\]

Error

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Results

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Target

Original14.5
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.5

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

  2. Initial simplification14.5

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

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

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

  5. Applied associate-/r*14.5

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

  7. Applied add-sqr-sqrt14.6

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

  8. Applied *-un-lft-identity14.6

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

  9. Applied times-frac14.5

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

  10. Simplified14.5

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Runtime

Time bar (total: 6.2s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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