Average Error: 3.3 → 1.5
Time: 6.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.193960641989379 \cdot 10^{+198}:\\ \;\;\;\;\left(\frac{\sqrt{1}}{\left(\beta + \alpha\right) + \left(1.0 + 2\right)} \cdot \frac{\sqrt{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}}{\left(2 + \alpha\right) + \beta}\right) \cdot \frac{\sqrt{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}}{\left(2 + \alpha\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.193960641989379e+198

    1. Initial program 1.6

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

    2. Applied simplify8.4

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

    4. Applied add-sqr-sqrt8.4

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

    5. Applied times-frac2.6

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

    7. Applied *-un-lft-identity2.6

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

    8. Applied sqrt-prod2.6

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

    9. Applied times-frac1.7

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

    if 4.193960641989379e+198 < beta

    1. Initial program 15.5

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

    2. Applied simplify15.5

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

    3. Taylor expanded around inf 0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 6.3m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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)))