Average Error: 45.5 → 0.0
Time: 22.4s
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 220.25806019550095:\\ \;\;\;\;\left(\frac{i}{2} \cdot \frac{i}{2}\right) \cdot \frac{1}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 220.25806019550095

    1. Initial program 44.4

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

    2. Initial simplification0.0

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

    4. Applied associate-*l/0.0

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

    5. Simplified0.0

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

    7. Applied div-inv0.0

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

    if 220.25806019550095 < i

    1. Initial program 46.5

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

    2. Initial simplification31.4

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

    4. Applied associate-*l/31.3

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

    5. Simplified31.3

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

    7. Applied div-inv31.3

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

    8. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    9. Simplified0.0

      \[\leadsto \color{blue}{(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 220.25806019550095:\\ \;\;\;\;\left(\frac{i}{2} \cdot \frac{i}{2}\right) \cdot \frac{1}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018230 +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)))