Average Error: 16.5 → 3.3
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 2.2717766713815923 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} \cdot \sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}{\sqrt{\sqrt{\left(\alpha + \beta\right) + 2.0}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} \cdot \sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}}{\sqrt{\sqrt{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

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 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 2.2717766713815923e-08

    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-58.0

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

    5. Taylor expanded around inf 10.9

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

    if 2.2717766713815923e-08 < (/ (+ (/ (- 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 add-sqr-sqrt0.2

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

    7. Applied associate-/r*0.2

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

    9. Applied add-sqr-sqrt0.2

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

    10. Applied sqrt-prod0.2

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

    11. Applied add-cube-cbrt0.4

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

    12. Applied times-frac0.4

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

    14. Applied add-cube-cbrt0.4

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

    15. Applied cbrt-prod0.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018208 +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))