Average Error: 3.3 → 1.4
Time: 6.7m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.2611144943335644 \cdot 10^{+194}:\\ \;\;\;\;\left(\frac{\sqrt{\sqrt{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}}}{\left(\beta + \alpha\right) + \left(1.0 + 2\right)} \cdot \frac{\sqrt{\sqrt{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}}}{\left(2 + \alpha\right) + \beta}\right) \cdot \frac{\sqrt{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}}{\left(2 + \alpha\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.2611144943335644e+194

    1. Initial program 1.5

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

    2. Applied simplify8.2

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

    4. Applied add-sqr-sqrt8.2

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

    5. Applied times-frac2.5

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

    7. Applied add-sqr-sqrt2.5

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

    8. Applied sqrt-prod2.5

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

    9. Applied times-frac1.6

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

    if 1.2611144943335644e+194 < alpha

    1. Initial program 16.6

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

    2. Applied simplify16.5

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

    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: 6.7m)Debug log

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