Average Error: 3.4 → 0.9
Time: 9.7m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.696184053442992 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\frac{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 + \left(\alpha + \beta\right)}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2 + \left(\alpha + \beta\right)}}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\beta}^{2}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.696184053442992e+161

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied associate-/l*1.1

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

    if 5.696184053442992e+161 < alpha

    1. Initial program 16.0

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

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

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

    4. Applied associate-/l*16.0

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

    6. Applied div-inv16.0

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

    7. Applied associate-/r*16.0

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

    8. Simplified16.0

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

    9. Taylor expanded around inf 0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.696184053442992 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\frac{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 + \left(\alpha + \beta\right)}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2 + \left(\alpha + \beta\right)}}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\beta}^{2}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 9.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes3.40.90.03.473%
herbie shell --seed 2018285 +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)))