Average Error: 45.6 → 0.3
Time: 19.3s
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{1}{16 - \frac{4.0}{i \cdot i}}\]

Error

Bits error versus i

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.6

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

  2. Initial simplification15.6

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

  4. Applied associate-*l/15.5

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

  5. Simplified15.5

    \[\leadsto \frac{\color{blue}{\frac{i}{2} \cdot \frac{i}{2}}}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*}\]
  6. Using strategy rm

  7. Applied clear-num15.8

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

  8. Taylor expanded around inf 0.3

    \[\leadsto \frac{1}{\color{blue}{16 - 4.0 \cdot \frac{1}{{i}^{2}}}}\]

  9. Simplified0.3

    \[\leadsto \frac{1}{\color{blue}{16 - \frac{4.0}{i \cdot i}}}\]

  10. Final simplification0.3

    \[\leadsto \frac{1}{16 - \frac{4.0}{i \cdot i}}\]

Runtime

Time bar (total: 19.3s)Debug logProfile

herbie shell --seed 2018225 +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)))