Average Error: 3.5 → 2.1
Time: 8.7m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.6198811786023372 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.6198811786023372e+162

    1. Initial program 1.2

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

    3. Applied associate-+l+1.2

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

    if 1.6198811786023372e+162 < alpha

    1. Initial program 16.2

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

    3. Applied associate-+l+16.2

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

    5. Applied *-un-lft-identity16.2

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

    6. Applied *-un-lft-identity16.2

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

    7. Applied distribute-lft-out16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Simplified16.2

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

    11. Simplified17.3

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

    12. Taylor expanded around 0 6.7

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

    13. Simplified6.7

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.6198811786023372 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 8.7m)Debug logProfile

herbie shell --seed 2018349 +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)))