Average Error: 16.5 → 3.2
Time: 7.8m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.2321443265605432 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\frac{\frac{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} \cdot {\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3} \cdot {\left({1.0}^{3}\right)}^{3}}{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}}{\left({1.0}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + {1.0}^{3} \cdot {1.0}^{3}\right) + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0 + 1.0 \cdot 1.0\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}}{2.0}\\ \end{array}\]

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 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 1.2321443265605432e-09

    1. Initial program 60.1

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

    3. Applied div-sub60.1

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

    4. Applied associate-+l-58.2

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

    5. 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 1.2321443265605432e-09 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    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 flip3--0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]
    7. Using strategy rm

    8. Applied flip3--0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{\frac{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {1.0}^{3}\right)}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]
    9. Using strategy rm

    10. Applied flip--0.2

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.2321443265605432 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\frac{\frac{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} \cdot {\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3} \cdot {\left({1.0}^{3}\right)}^{3}}{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}}{\left({1.0}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + {1.0}^{3} \cdot {1.0}^{3}\right) + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0 + 1.0 \cdot 1.0\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 7.8m)Debug logProfile

herbie shell --seed 2018217 
(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))