Average Error: 3.7 → 2.3
Time: 6.4m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.7336741476979517 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \left(1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)\right)}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.7336741476979517e+164

    1. Initial program 1.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied div-inv1.3

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

    if 1.7336741476979517e+164 < alpha

    1. Initial program 17.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied div-inv17.3

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.3

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

    6. Applied *-un-lft-identity17.3

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

    7. Applied times-frac17.3

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

    8. Simplified17.3

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

    9. Simplified18.2

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

    10. Taylor expanded around 0 7.9

      \[\leadsto 1 \cdot \frac{\color{blue}{0.5 + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\]

    11. Simplified7.9

      \[\leadsto 1 \cdot \frac{\color{blue}{(\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*}}{\left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + 2\right) + \alpha\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.7336741476979517 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \left(1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)\right)}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))