Average Error: 45.8 → 0.4
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\log_* (1 + (e^{\frac{1}{16 - \frac{4.0}{i \cdot i}}} - 1)^*)\]

Error

Bits error versus i

Try it out

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. Initial simplification15.8

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

  4. Applied log1p-expm1-u15.8

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{i \cdot \frac{i}{4}}{(\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}} - 1)^*)}\]
  5. Using strategy rm

  6. Applied clear-num16.2

    \[\leadsto \log_* (1 + (e^{\color{blue}{\frac{1}{\frac{(\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}{i \cdot \frac{i}{4}}}}} - 1)^*)\]

  7. Taylor expanded around -inf 0.4

    \[\leadsto \log_* (1 + (e^{\frac{1}{\color{blue}{16 - 4.0 \cdot \frac{1}{{i}^{2}}}}} - 1)^*)\]

  8. Simplified0.4

    \[\leadsto \log_* (1 + (e^{\frac{1}{\color{blue}{16 - \frac{4.0}{i \cdot i}}}} - 1)^*)\]

  9. Final simplification0.4

    \[\leadsto \log_* (1 + (e^{\frac{1}{16 - \frac{4.0}{i \cdot i}}} - 1)^*)\]

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed 2018346 +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)))