Average Error: 52.6 → 31.3
Time: 6.0m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 6.691590134286189 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i + \left(\beta + \alpha\right)}}}{\frac{\sqrt{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}}{\left(i + \alpha\right) + \beta}}}{\frac{\sqrt{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}}{\frac{1}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i + \left(\beta + \alpha\right)}}}{(7 \cdot i + \left(\alpha \cdot 5 - \frac{1.0}{i}\right))_*}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 6.691590134286189e+153

    1. Initial program 42.9

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

    2. Simplified42.9

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

    4. Applied associate-/l*15.3

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

    5. Taylor expanded around inf 15.7

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

    6. Simplified15.7

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

    8. Applied times-frac15.7

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

    9. Applied times-frac11.1

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

    10. Applied associate-/l*11.1

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

    12. Applied div-inv11.2

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

    13. Applied add-sqr-sqrt11.2

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

    14. Applied times-frac11.1

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

    15. Applied associate-/r*11.1

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

    if 6.691590134286189e+153 < 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. Simplified62.1

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

    4. Applied associate-/l*62.1

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

    5. Taylor expanded around inf 62.1

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

    6. Simplified62.1

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

    8. Applied times-frac62.1

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

    9. Applied times-frac62.1

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

    10. Applied associate-/l*61.9

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

    11. Taylor expanded around 0 50.9

      \[\leadsto \frac{\frac{i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{\left(\alpha + \beta\right) + i}}}{\color{blue}{\left(7 \cdot i + 5 \cdot \alpha\right) - 1.0 \cdot \frac{1}{i}}}\]

    12. Simplified50.9

      \[\leadsto \frac{\frac{i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{\left(\alpha + \beta\right) + i}}}{\color{blue}{(7 \cdot i + \left(5 \cdot \alpha - \frac{1.0}{i}\right))_*}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.691590134286189 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i + \left(\beta + \alpha\right)}}}{\frac{\sqrt{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}}{\left(i + \alpha\right) + \beta}}}{\frac{\sqrt{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}}{\frac{1}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_*}{i + \left(\beta + \alpha\right)}}}{(7 \cdot i + \left(\alpha \cdot 5 - \frac{1.0}{i}\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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)))