Average Error: 52.3 → 12.3
Time: 2.4m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1.144345765286438 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}}{\frac{1}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\beta + \alpha\right) + i\right)}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) \cdot \left(2 \cdot i + \left(\beta + \alpha\right)\right) - 1.0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}} \cdot \left(\sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}} \cdot \sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}}\right)}{4}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1.144345765286438e+119

    1. Initial program 36.7

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

    3. Applied associate-/l*13.2

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

    5. Applied associate-/r/13.2

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

    6. Applied associate-/l*13.2

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

    7. Simplified13.2

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

    9. Applied *-un-lft-identity13.2

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

    10. Applied associate-/l*13.2

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

    if 1.144345765286438e+119 < 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*54.7

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

    5. Applied associate-/r/54.7

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

    6. Applied associate-/l*54.7

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

    7. Simplified54.7

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

    8. Taylor expanded around 0 11.8

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

    10. Applied add-cube-cbrt11.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{i}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \alpha)_* + \beta}} \cdot \sqrt[3]{\frac{i}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \alpha)_* + \beta}}\right) \cdot \sqrt[3]{\frac{i}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \alpha)_* + \beta}}}}{4}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.144345765286438 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}}{\frac{1}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\beta + \alpha\right) + i\right)}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) \cdot \left(2 \cdot i + \left(\beta + \alpha\right)\right) - 1.0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}} \cdot \left(\sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}} \cdot \sqrt[3]{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}}\right)}{4}\\ \end{array}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes14.412.38.55.935%
herbie shell --seed 2018290 +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)))