Average Error: 52.6 → 12.8
Time: 7.7m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 9.964308556238813 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\alpha + \left(\beta + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\alpha + \left(\beta + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\alpha + \left(\beta + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\alpha + \left(\beta + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{0.25}{i}}{i} + \log \frac{1}{16}}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 9.964308556238813e+149

    1. Initial program 42.6

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    2. Using strategy rm

    3. Applied associate-/l*15.7

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt15.7

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

    6. Applied add-sqr-sqrt15.7

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

    7. Applied times-frac15.7

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

    8. Simplified15.7

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

    9. Simplified15.7

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

    if 9.964308556238813e+149 < i

    1. Initial program 62.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    2. Using strategy rm

    3. Applied associate-/l*60.9

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    4. Using strategy rm

    5. Applied add-exp-log60.9

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

    6. Taylor expanded around inf 10.1

      \[\leadsto e^{\color{blue}{\log \frac{1}{16} + 0.25 \cdot \frac{1}{{i}^{2}}}}\]

    7. Simplified10.1

      \[\leadsto e^{\color{blue}{\log \frac{1}{16} + \frac{\frac{0.25}{i}}{i}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 9.964308556238813 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\alpha + \left(\beta + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\alpha + \left(\beta + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\alpha + \left(\beta + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\alpha + \left(\beta + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{0.25}{i}}{i} + \log \frac{1}{16}}\\ \end{array}\]

Runtime

Time bar (total: 7.7m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))