Average Error: 46.1 → 0.2
Time: 46.6s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{2}}{8 \cdot i - \frac{2.0}{i}}\]

Error

Bits error versus i

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.1

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

  2. Initial simplification15.4

    \[\leadsto \frac{\frac{i}{\frac{2}{1}}}{\frac{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*}{\frac{i}{\frac{2}{1}}}}\]

  3. Taylor expanded around 0 0.2

    \[\leadsto \frac{\frac{i}{\frac{2}{1}}}{\color{blue}{8 \cdot i - 2.0 \cdot \frac{1}{i}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\frac{i}{\frac{2}{1}}}{\color{blue}{i \cdot 8 - \frac{2.0}{i}}}\]

  5. Final simplification0.2

    \[\leadsto \frac{\frac{i}{2}}{8 \cdot i - \frac{2.0}{i}}\]

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))