Average Error: 45.6 → 0.0
Time: 3.7m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 220.44579005536056:\\ \;\;\;\;\log_* (1 + (e^{\frac{\frac{i}{2} \cdot \frac{i}{2}}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 220.44579005536056

    1. Initial program 43.9

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

    2. Applied simplify0.2

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

    4. Applied log1p-expm1-u0.2

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

    5. Applied simplify0.0

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

    if 220.44579005536056 < i

    1. Initial program 47.2

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

    2. Applied simplify31.5

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

    3. Taylor expanded around inf 0.0

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

    4. Applied simplify0

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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)))