Average Error: 45.8 → 0.2
Time: 6.3s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{i}{i \cdot 16 - \frac{4.0}{i}}\]

Error

Bits error versus i

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.8

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

  2. Simplified15.2

    \[\leadsto \color{blue}{\frac{i \cdot \frac{i}{4}}{(\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}}\]
  3. Using strategy rm

  4. Applied associate-/l*15.3

    \[\leadsto \color{blue}{\frac{i}{\frac{(\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}{\frac{i}{4}}}}\]

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019030 +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)))