Average Error: 15.9 → 5.8
Time: 23.4s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 16268357.0012352:\\ \;\;\;\;\frac{\log \left(e^{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta}\right) - \left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2.0}{\alpha}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 16268357.0012352

    1. Initial program 0.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied clear-num0.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\alpha}}} - 1.0\right)}{2.0}\]
    7. Using strategy rm

    8. Applied div-inv0.1

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\alpha}} - 1.0\right)}{2.0}\]
    9. Using strategy rm

    10. Applied add-log-exp0.1

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

    if 16268357.0012352 < alpha

    1. Initial program 49.4

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub49.4

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

    4. Applied associate-+l-48.0

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied clear-num48.0

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\alpha}}} - 1.0\right)}{2.0}\]
    7. Using strategy rm

    8. Applied div-inv48.0

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

    9. Taylor expanded around -inf 17.9

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    10. Simplified17.9

      \[\leadsto \frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 16268357.0012352:\\ \;\;\;\;\frac{\log \left(e^{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta}\right) - \left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2.0}{\alpha}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \frac{\left(\frac{4.0}{\alpha} - 2.0\right) - \frac{8.0}{\alpha \cdot \alpha}}{\alpha}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 23.4s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))