Average Error: 52.6 → 8.5
Time: 6.4m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8} \le 1.469376147118379 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{\left(\frac{\alpha}{\beta} + 2\right) + \frac{\beta}{\alpha}}\\ \mathbf{elif}\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8} \le 0.062482464860213256:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{\left(\frac{(i \cdot 2 + \beta)_* + \alpha}{\left(i + \alpha\right) + \beta} \cdot \left((i \cdot 2 + \beta)_* + \alpha\right)\right) \cdot \frac{(i \cdot 2 + \beta)_* + \alpha}{(i \cdot \left(\left(i + \alpha\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*} - \frac{1.0}{\frac{\left(\beta + \alpha\right) + i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i \cdot \left(\left(\beta + \alpha\right) + i\right) + \alpha \cdot \beta}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (/ (/ i (+ (fma i 2 beta) alpha)) 8) < 1.469376147118379e-142

    1. Initial program 62.7

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

    3. Applied associate-/l*42.8

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

    5. Applied *-un-lft-identity42.8

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

    6. Applied times-frac42.8

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

    7. Applied times-frac42.8

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

    8. Applied associate-/l*42.8

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

    9. Simplified42.8

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

    10. Taylor expanded around inf 9.7

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

    if 1.469376147118379e-142 < (/ (/ i (+ (fma i 2 beta) alpha)) 8) < 0.062482464860213256

    1. Initial program 54.0

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

    3. Applied associate-/l*42.3

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

    5. Applied *-un-lft-identity42.3

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

    6. Applied times-frac42.3

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

    7. Applied times-frac42.3

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

    8. Applied associate-/l*42.3

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

    9. Simplified42.3

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

    11. Applied div-sub42.3

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

    12. Simplified33.7

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

    if 0.062482464860213256 < (/ (/ i (+ (fma i 2 beta) alpha)) 8)

    1. Initial program 50.7

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

    3. Applied associate-/l*36.5

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

    5. Applied *-un-lft-identity36.5

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

    6. Applied times-frac36.7

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

    7. Applied times-frac36.6

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

    8. Applied associate-/l*36.6

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

    9. Simplified36.6

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

    10. Taylor expanded around 0 0.9

      \[\leadsto \frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{\color{blue}{8}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8} \le 1.469376147118379 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{\left(\frac{\alpha}{\beta} + 2\right) + \frac{\beta}{\alpha}}\\ \mathbf{elif}\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8} \le 0.062482464860213256:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{\left(\frac{(i \cdot 2 + \beta)_* + \alpha}{\left(i + \alpha\right) + \beta} \cdot \left((i \cdot 2 + \beta)_* + \alpha\right)\right) \cdot \frac{(i \cdot 2 + \beta)_* + \alpha}{(i \cdot \left(\left(i + \alpha\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*} - \frac{1.0}{\frac{\left(\beta + \alpha\right) + i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i \cdot \left(\left(\beta + \alpha\right) + i\right) + \alpha \cdot \beta}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{(i \cdot 2 + \beta)_* + \alpha}}{8}\\ \end{array}\]

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed 2018217 +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)))