Average Error: 15.9 → 7.0
Time: 21.8s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 172751.08339392173:\\ \;\;\;\;\frac{\frac{\frac{\beta}{e^{\log \left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}}}{{\left(2.0 + \left(\beta + \alpha\right)\right)}^{\frac{1}{3}}} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta}{e^{\log \left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 172751.08339392173

    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-cube-cbrt0.3

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

    7. Applied associate-/r*0.3

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

    9. Applied add-exp-log1.2

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

    11. Applied pow1/31.2

      \[\leadsto \frac{\frac{\frac{\beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot e^{\log \left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right)}}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    if 172751.08339392173 < alpha

    1. Initial program 49.1

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

    3. Applied div-sub49.0

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

    4. Applied associate-+l-47.4

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

    6. Applied add-cube-cbrt47.6

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

    7. Applied associate-/r*47.6

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

    9. Applied add-exp-log48.1

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

    10. Taylor expanded around -inf 19.0

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

    11. Simplified19.0

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 172751.08339392173:\\ \;\;\;\;\frac{\frac{\frac{\beta}{e^{\log \left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}}}{{\left(2.0 + \left(\beta + \alpha\right)\right)}^{\frac{1}{3}}} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta}{e^{\log \left(\sqrt[3]{2.0 + \left(\beta + \alpha\right)}\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 21.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes16.47.03.213.271.4%
herbie shell --seed 2018295 +o rules:numerics
(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))