Average Error: 16.4 → 0.2
Time: 50.9s
Precision: 64
Internal precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
⬇
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999988313739:\\ \;\;\;\;\frac{\frac{8.0}{{\alpha}^2} + 2.0}{2.0 \cdot \alpha} + \left(\frac{\frac{\beta}{2.0}}{\left(2.0 + \alpha\right) + \beta} - \frac{\frac{4.0}{{\alpha}^2}}{2.0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\sqrt[3]{{\left({\left(\frac{\alpha}{\beta + \left(2.0 + \alpha\right)}\right)}^3\right)}^3} - {1.0}^3}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes.

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999988313739

    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-sub 60.0

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

    4. Applied associate-+l- 49.1

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

    5. Applied taylor 0.0

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

    6. Taylor expanded around inf 0.0

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

    7. Applied simplify 0.1

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

    if -0.9999999988313739 < (/ (- beta alpha) (+ (+ alpha beta) 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-sub 0.2

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

    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

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]

    7. Applied simplify 0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{{\left(\frac{\alpha}{\beta + \left(2.0 + \alpha\right)}\right)}^3 - {1.0}^3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube 0.2

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

Runtime

Time bar (total: 50.9s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2703974819 3200048092 2412161911 273771099 3921942679 3608474975)'
(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))