Average Error: 16.4 → 3.1
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha} \le -1.0000006494352762:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}\right)}{2.0}\\ \mathbf{if}\;\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha} \le -1.0:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right) + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{-4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\\ \end{array}\]

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 3 regimes

  2. if (/ (+ (+ alpha beta) 2.0) (- beta alpha)) < -1.0000006494352762

    1. Initial program 0.3

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

    3. Applied add-log-exp0.3

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

    if -1.0000006494352762 < (/ (+ (+ alpha beta) 2.0) (- beta alpha)) < -1.0

    1. Initial program 59.8

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

    3. Applied div-sub59.8

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

    4. Applied associate-+l-57.9

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

    6. Applied add-sqr-sqrt57.9

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

    7. Applied associate-/r*57.9

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

    8. Taylor expanded around inf 11.3

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

    9. Applied simplify11.3

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

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

    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 clear-num0.0

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(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))