Average Error: 15.8 → 0.1
Time: 3.2s
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.8
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 15.8

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

  2. Initial simplification15.8

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

  4. Applied clear-num15.9

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

  5. Taylor expanded around inf 0.1

    \[\leadsto \frac{1}{\color{blue}{x + \frac{1}{x}}}\]

  6. Final simplification0.1

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

Runtime

Time bar (total: 3.2s)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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