Average Error: 15.4 → 0.2
Time: 20.4s
Precision: 64
Internal Precision: 320
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.003492093787603 \lor \neg \left(x \le 0.9954769736658148\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} + x\right) - {x}^{3}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.4
Target0.1
Herbie0.2
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.003492093787603 or 0.9954769736658148 < x

    1. Initial program 30.2

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

    2. Initial simplification30.2

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

    3. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]

    if -1.003492093787603 < x < 0.9954769736658148

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + {x}^{5}\right) - {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.003492093787603 \lor \neg \left(x \le 0.9954769736658148\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} + x\right) - {x}^{3}\\ \end{array}\]

Runtime

Time bar (total: 20.4s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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