Average Error: 46.5 → 0.0
Time: 7.8s
Precision: 64
Internal precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
⬇
\[\begin{array}{l} \mathbf{if}\;i \le 14343.200142683158:\\ \;\;\;\;\frac{\frac{i}{2} \cdot \frac{i}{2}}{{\left(i + i\right)}^2 - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{{i}^2} + \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right)\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes.

  2. if i < 14343.200142683158

    1. Initial program 44.6

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

    2. Applied simplify 0.0

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

    3. Applied simplify 0.0

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

    if 14343.200142683158 < i

    1. Initial program 48.5

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

    2. Applied simplify 32.8

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

    3. Applied taylor 0

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

    4. Taylor expanded around inf 0

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

    5. Applied simplify 0

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

    6. Applied simplify 0

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

Runtime

Time bar (total: 7.8s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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)))