Average Error: 3.8 → 1.7
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \cdot \alpha \le 3.509388025551998 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{\sqrt{\left(\beta + 2\right) + \alpha}}}{\sqrt{\left(\alpha + 1.0\right) + \left(\beta + 2\right)} \cdot \left(\left(\beta + 2\right) + \alpha\right)} \cdot \frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{\sqrt{\left(\beta + 1.0\right) + \left(2 + \alpha\right)} \cdot \sqrt{\left(\beta + 2\right) + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{6.0}{\alpha}}{\alpha} + \left(1 - \frac{2.0}{\alpha}\right)}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \end{array}\]

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 (* beta alpha) < 3.509388025551998e+299

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.6

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

    4. Applied *-un-lft-identity0.6

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

    5. Applied add-sqr-sqrt0.3

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

    6. Applied add-sqr-sqrt0.2

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

    7. Applied times-frac0.2

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

    8. Applied times-frac0.2

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

    9. Applied times-frac0.2

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

    10. Simplified0.3

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

    11. Simplified0.2

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

    if 3.509388025551998e+299 < (* beta alpha)

    1. Initial program 58.8

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

    2. Initial simplification60.8

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

    3. Taylor expanded around inf 24.1

      \[\leadsto \frac{\color{blue}{\left(6.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 2.0 \cdot \frac{1}{\alpha}}}{\left(\left(\alpha + 2\right) + \beta\right) \cdot \left(\left(\alpha + 2\right) + \beta\right)}\]

    4. Simplified24.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{6.0}{\alpha}}{\alpha} + \left(1 - \frac{2.0}{\alpha}\right)}}{\left(\left(\alpha + 2\right) + \beta\right) \cdot \left(\left(\alpha + 2\right) + \beta\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \cdot \alpha \le 3.509388025551998 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{\sqrt{\left(\beta + 2\right) + \alpha}}}{\sqrt{\left(\alpha + 1.0\right) + \left(\beta + 2\right)} \cdot \left(\left(\beta + 2\right) + \alpha\right)} \cdot \frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{\sqrt{\left(\beta + 1.0\right) + \left(2 + \alpha\right)} \cdot \sqrt{\left(\beta + 2\right) + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{6.0}{\alpha}}{\alpha} + \left(1 - \frac{2.0}{\alpha}\right)}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \end{array}\]

Runtime

Time bar (total: 5.2m)Debug logProfile

herbie shell --seed 2018277 
(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)))