Error

Derivation

1. Split input into 2 regimes

2. if (+ alpha beta) < 1.6844177380724597e+188

   1. Initial program 0.3

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

   3. Applied *-un-lft-identity0.3

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

   4. Applied add-sqr-sqrt0.5

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

   5. Applied times-frac0.5

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

   6. Applied simplify0.5

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

   if 1.6844177380724597e+188 < (+ alpha beta)

   1. Initial program 13.4

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

   2. Taylor expanded around 0 59.0

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

   3. Applied simplify4.1

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

4. Applied simplify1.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\beta + \alpha \le 1.6844177380724597 \cdot 10^{+188}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\beta + 1.0\right) + \alpha\right) + \beta \cdot \alpha}}{2 + \left(\beta + \alpha\right)} \cdot \frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1.0}}{1}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\beta + \alpha\right) + 0.5}{\left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 9.5m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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)))