Average Error: 3.7 → 1.7
Time: 6.9m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta + \alpha \le 6.954326581165607 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\beta + \beta \cdot \alpha\right) + \left(\alpha + 1.0\right)}{\left(\beta + \alpha\right) + 2}\right)}^{3}}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (+ alpha beta) < 6.954326581165607e+191

    1. Initial program 0.5

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

    3. Applied add-cbrt-cube8.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    4. Applied add-cbrt-cube13.2

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    5. Applied cbrt-undiv13.2

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied simplify0.9

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(\frac{\left(\alpha + 1.0\right) + \left(\beta \cdot \alpha + \beta\right)}{\left(\beta + \alpha\right) + 2}\right)}^{3}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    if 6.954326581165607e+191 < (+ alpha beta)

    1. Initial program 13.8

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

    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 simplify1.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\beta + \alpha \le 6.954326581165607 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\beta + \beta \cdot \alpha\right) + \left(\alpha + 1.0\right)}{\left(\beta + \alpha\right) + 2}\right)}^{3}}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}}\]

Runtime

Time bar (total: 6.9m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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)))