Average Error: 3.6 → 2.6
Time: 5.5m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.4204792506782105 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{(\alpha \cdot \beta + \beta)_* + \left(\alpha + 1.0\right)}}{\sqrt{\left(2 + \alpha\right) + \beta}} \cdot \frac{\sqrt{(\alpha \cdot \beta + \beta)_* + \left(\alpha + 1.0\right)}}{\sqrt{\left(2 + \alpha\right) + \beta}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(\beta + 2\right) + \left(\alpha + 1.0\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.4204792506782105e+171

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.9

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

    4. Applied add-sqr-sqrt1.8

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

    5. Applied times-frac1.8

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

    6. Applied simplify1.8

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

    7. Applied simplify1.8

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

    if 5.4204792506782105e+171 < 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. Taylor expanded around inf 6.8

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

    3. Applied simplify6.8

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

  4. Applied simplify2.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\alpha \le 5.4204792506782105 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{(\alpha \cdot \beta + \beta)_* + \left(\alpha + 1.0\right)}}{\sqrt{\left(2 + \alpha\right) + \beta}} \cdot \frac{\sqrt{(\alpha \cdot \beta + \beta)_* + \left(\alpha + 1.0\right)}}{\sqrt{\left(2 + \alpha\right) + \beta}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(\beta + 2\right) + \left(\alpha + 1.0\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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)))