Average Error: 45.7 → 0.3
Time: 4.3s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 0.5058304269271637:\\ \;\;\;\;\left(-4.0\right) \cdot {i}^{6} - (\left(i \cdot 0.25\right) \cdot i + \left({i}^{4} \cdot 1.0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 0.5058304269271637

    1. Initial program 44.8

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

    2. Initial simplification0.0

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

    3. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{-\left(0.25 \cdot {i}^{2} + \left(1.0 \cdot {i}^{4} + 4.0 \cdot {i}^{6}\right)\right)}\]

    4. Simplified0.3

      \[\leadsto \color{blue}{{i}^{6} \cdot \left(-4.0\right) - (\left(0.25 \cdot i\right) \cdot i + \left({i}^{4} \cdot 1.0\right))_*}\]

    if 0.5058304269271637 < i

    1. Initial program 46.7

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

    2. Initial simplification31.7

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 0.5058304269271637:\\ \;\;\;\;\left(-4.0\right) \cdot {i}^{6} - (\left(i \cdot 0.25\right) \cdot i + \left({i}^{4} \cdot 1.0\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}\\ \end{array}\]

Runtime

Time bar (total: 4.3s)Debug logProfile

herbie shell --seed 2018254 +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)))