Average Error: 3.6 → 2.1
Time: 6.8m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \le 0.08598267559468141:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\left(\alpha \cdot \beta + \left(\beta + 1.0\right)\right) + \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right)} \cdot \left(\sqrt[3]{\frac{\alpha \cdot \beta + \left(\left(\beta + \alpha\right) + 1.0\right)}{\left(\alpha + 2\right) + \beta}} \cdot \frac{\sqrt[3]{\frac{\alpha \cdot \beta + \left(\left(\beta + \alpha\right) + 1.0\right)}{\left(\alpha + 2\right) + \beta}}}{\sqrt{\left(\alpha + 2\right) + \beta}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2.0}{\alpha}}{\alpha} + \left(1 - \frac{1.0}{\alpha}\right)}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

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 beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 0.08598267559468141

    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

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

    4. Applied add-sqr-sqrt0.6

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

    5. Applied add-cube-cbrt0.3

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}\right) \cdot \sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]

    6. Applied times-frac0.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]

    7. Applied times-frac0.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}}\]

    8. Simplified0.7

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{\alpha \cdot \beta + \left(1.0 + \left(\beta + \alpha\right)\right)}{\left(\alpha + 2\right) + \beta}}}{\sqrt{\left(\alpha + 2\right) + \beta}} \cdot \sqrt[3]{\frac{\alpha \cdot \beta + \left(1.0 + \left(\beta + \alpha\right)\right)}{\left(\alpha + 2\right) + \beta}}\right)} \cdot \frac{\frac{\sqrt[3]{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    9. Simplified0.7

      \[\leadsto \left(\frac{\sqrt[3]{\frac{\alpha \cdot \beta + \left(1.0 + \left(\beta + \alpha\right)\right)}{\left(\alpha + 2\right) + \beta}}}{\sqrt{\left(\alpha + 2\right) + \beta}} \cdot \sqrt[3]{\frac{\alpha \cdot \beta + \left(1.0 + \left(\beta + \alpha\right)\right)}{\left(\alpha + 2\right) + \beta}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{\left(\left(\beta + 1.0\right) + \alpha \cdot \beta\right) + \alpha}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right)}}\]

    if 0.08598267559468141 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 61.1

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

    2. Taylor expanded around inf 24.4

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

    3. Simplified24.4

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \le 0.08598267559468141:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\left(\alpha \cdot \beta + \left(\beta + 1.0\right)\right) + \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right)} \cdot \left(\sqrt[3]{\frac{\alpha \cdot \beta + \left(\left(\beta + \alpha\right) + 1.0\right)}{\left(\alpha + 2\right) + \beta}} \cdot \frac{\sqrt[3]{\frac{\alpha \cdot \beta + \left(\left(\beta + \alpha\right) + 1.0\right)}{\left(\alpha + 2\right) + \beta}}}{\sqrt{\left(\alpha + 2\right) + \beta}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2.0}{\alpha}}{\alpha} + \left(1 - \frac{1.0}{\alpha}\right)}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 6.8m)Debug logProfile

herbie shell --seed 2018221 
(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)))