Average Error: 3.9 → 2.8
Time: 5.8m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1147026063016264 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\beta + 2\right) + \alpha}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\beta}\right) \cdot \left(\frac{2.0}{\beta} - 1.0\right) + 1)_* \cdot \frac{1}{\left(\beta + 2\right) + \alpha}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.1147026063016264e+125

    1. Initial program 0.8

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

    3. Applied add-sqr-sqrt1.4

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

    4. Applied add-sqr-sqrt1.8

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

    5. Applied *-un-lft-identity1.8

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

    6. Applied times-frac1.8

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

    7. Applied times-frac1.6

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

    8. Simplified1.0

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

    9. Simplified0.9

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

    if 1.1147026063016264e+125 < beta

    1. Initial program 15.9

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

    3. Applied add-sqr-sqrt15.9

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

    4. Applied add-sqr-sqrt16.0

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

    5. Applied *-un-lft-identity16.0

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

    6. Applied times-frac16.0

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

    7. Applied times-frac16.0

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

    8. Simplified16.0

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

    9. Simplified15.9

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

    10. Taylor expanded around -inf 10.3

      \[\leadsto \frac{\frac{1}{\left(\beta + 2\right) + \alpha} \cdot \color{blue}{\left(\left(2.0 \cdot \frac{1}{{\beta}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    11. Simplified10.3

      \[\leadsto \frac{\frac{1}{\left(\beta + 2\right) + \alpha} \cdot \color{blue}{(\left(\frac{1}{\beta}\right) \cdot \left(\frac{2.0}{\beta} - 1.0\right) + 1)_*}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1147026063016264 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\beta + 2\right) + \alpha}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\beta}\right) \cdot \left(\frac{2.0}{\beta} - 1.0\right) + 1)_* \cdot \frac{1}{\left(\beta + 2\right) + \alpha}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(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)))