Error

Derivation

1. Split input into 2 regimes.

2. if alpha < 1.4121013280836876e+163

   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. Applied simplify 2.2

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

   4. Applied *-un-lft-identity 2.2

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

   5. Applied *-un-lft-identity 2.2

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

   6. Applied times-frac 2.2

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

   7. Applied times-frac 1.4

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

   8. Applied simplify 1.4

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

   if 1.4121013280836876e+163 < alpha

   1. Initial program 60.3

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

   2. Applied simplify 60.3

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

   3. Applied taylor 59.3

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

   4. Taylor expanded around 0 59.3

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

   5. Applied simplify 59.3

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\beta + \alpha\right) + 0.5}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(1.0 + \alpha\right) + \left(2 + \beta\right)\right)}}\]
3. Recombined 2 regimes into one program.
4. Removed slow pow expressions

Runtime

Time bar (total: 1.6m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2269281169 2100121522 1918668845 2444134882 3707986529 2154161493)'
(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)))