Average Error: 16.1 → 3.2
Time: 48.4s
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 1.3644640972643174 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\beta}{2.0}}{\beta + \left(2.0 + \alpha\right)} - \left(\frac{4.0 - \frac{8.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{1}{\alpha}}{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 1.3644640972643174e-13

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.4

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

    5. Taylor expanded around inf 11.3

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

    6. Applied simplify11.3

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

    if 1.3644640972643174e-13 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    1. Initial program 0.4

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied div-inv0.3

      \[\leadsto \frac{\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 48.4s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))