Average Error: 3.5 → 2.5
Time: 4.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.7232809148582944 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}}{\left(\alpha + \beta\right) + 2} \cdot \frac{\sqrt{(\alpha \cdot \beta + \alpha)_* + \left(\beta + 1.0\right)}}{1}}{\left(\alpha + \beta\right) + 2}}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2.0}{\alpha}}{\alpha} + \left(1 - \frac{1.0}{\alpha}\right)}{\left(\left(2 + \alpha\right) + \left(\beta + 1.0\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.7232809148582944e+134

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied add-sqr-sqrt1.0

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

    5. Applied times-frac1.0

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

    6. Applied simplify1.0

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

    7. Applied simplify1.0

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

    if 1.7232809148582944e+134 < alpha

    1. Initial program 15.1

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

    2. Taylor expanded around inf 8.4

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

    3. Applied simplify9.1

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

  4. Applied simplify2.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\alpha \le 1.7232809148582944 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}}{\left(\alpha + \beta\right) + 2} \cdot \frac{\sqrt{(\alpha \cdot \beta + \alpha)_* + \left(\beta + 1.0\right)}}{1}}{\left(\alpha + \beta\right) + 2}}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2.0}{\alpha}}{\alpha} + \left(1 - \frac{1.0}{\alpha}\right)}{\left(\left(2 + \alpha\right) + \left(\beta + 1.0\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

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