Average Error: 3.7 → 2.7
Time: 4.8m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.329325484181234 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}}{\left(\beta + \alpha\right) + 2} \cdot \frac{\left(\alpha + \left(\beta + 1.0\right)\right) + \beta \cdot \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\beta + \alpha\right) \cdot 0.25 + 0.5\right) \cdot \frac{\frac{1}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array}\]

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 < 6.329325484181234e+161

    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 *-un-lft-identity1.3

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

    4. Applied div-inv1.3

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

    5. Applied times-frac1.7

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

    6. Simplified1.7

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

    7. Simplified1.7

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

    if 6.329325484181234e+161 < alpha

    1. Initial program 16.3

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

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

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

    4. Applied div-inv16.3

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

    5. Applied times-frac17.6

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

    6. Simplified17.6

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

    7. Simplified17.6

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

    8. Taylor expanded around 0 7.8

      \[\leadsto \color{blue}{\left(0.25 \cdot \alpha + \left(0.25 \cdot \beta + 0.5\right)\right)} \cdot \frac{\frac{1}{\left(\alpha + 2\right) + \left(1.0 + \beta\right)}}{\left(\alpha + \beta\right) + 2}\]

    9. Simplified7.8

      \[\leadsto \color{blue}{\left(0.5 + 0.25 \cdot \left(\alpha + \beta\right)\right)} \cdot \frac{\frac{1}{\left(\alpha + 2\right) + \left(1.0 + \beta\right)}}{\left(\alpha + \beta\right) + 2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.329325484181234 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}}{\left(\beta + \alpha\right) + 2} \cdot \frac{\left(\alpha + \left(\beta + 1.0\right)\right) + \beta \cdot \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\beta + \alpha\right) \cdot 0.25 + 0.5\right) \cdot \frac{\frac{1}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array}\]

Runtime

Time bar (total: 4.8m)Debug logProfile

herbie shell --seed 2018234 
(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)))