Average Error: 16.2 → 3.2
Time: 29.9s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.9938043602110306 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{elif}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 0.5000001989457653:\\ \;\;\;\;\frac{\frac{\beta}{e^{\log \left(\left(\alpha + \beta\right) + 2.0\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} \cdot \sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

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 3 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 1.9938043602110306e-09

    1. Initial program 60.0

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

    3. Applied div-sub60.0

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

    4. Applied associate-+l-58.1

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

    5. Taylor expanded around inf 11.1

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

    6. Simplified11.1

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

    if 1.9938043602110306e-09 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 0.5000001989457653

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied add-exp-log31.2

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

    8. Applied log-div31.2

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

    9. Applied exp-diff31.2

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

    10. Simplified0.2

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

    if 0.5000001989457653 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied add-exp-log0.4

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

    8. Applied add-cube-cbrt0.4

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.9938043602110306 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{elif}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 0.5000001989457653:\\ \;\;\;\;\frac{\frac{\beta}{e^{\log \left(\left(\alpha + \beta\right) + 2.0\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} \cdot \sqrt[3]{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 29.9s)Debug logProfile

herbie shell --seed 2018216 
(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))