Average Error: 14.8 → 0.1
Time: 2.9s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.8

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

  2. Simplified14.8

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

  4. Applied clear-num14.8

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

  5. Taylor expanded around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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