Average Error: 3.3 → 1.7
Time: 8.4m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 2.709401827428557 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)} \cdot \frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{1}{\beta} - \frac{1}{\alpha}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{1}{\beta}}}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}\\ \end{array}\]

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 (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 2.709401827428557e+269

    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 *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.2

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

    5. Applied times-frac0.2

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

    6. Applied simplify0.2

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

    7. Applied simplify0.2

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

    if 2.709401827428557e+269 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 63.0

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

    3. Applied add-exp-log63.0

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

    4. Applied add-exp-log63.0

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

    5. Applied div-exp63.0

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

    6. Applied simplify63.0

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

    7. Taylor expanded around inf 30.0

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

    8. Applied simplify29.2

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

  4. Applied simplify1.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 2.709401827428557 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)} \cdot \frac{\sqrt{\beta \cdot \alpha + \left(\left(\beta + 1.0\right) + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{1}{\beta} - \frac{1}{\alpha}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{1}{\beta}}}{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}\\ \end{array}}\]

Runtime

Time bar (total: 8.4m)Debug logProfile

herbie shell --seed 2018178 
(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)))