Average Error: 15.9 → 3.0
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999847547237:\\ \;\;\;\;\frac{\frac{\beta}{2.0}}{2.0 + \left(\alpha + \beta\right)} - \frac{\frac{12.0 - \frac{32.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{4.0}{\alpha}}{2.0}}{1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\frac{{\left(\frac{\alpha}{\left(\alpha + 2.0\right) + \beta}\right)}^{\left(\left(1 + \left(1 + 3\right)\right) + 1\right)} - {\left(1.0 \cdot 1.0\right)}^{3}}{\left(\frac{\alpha \cdot 1.0}{\left(\alpha + 2.0\right) + \beta} \cdot \frac{\alpha \cdot 1.0}{\left(\alpha + 2.0\right) + \beta} + \left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + {\left(\frac{\alpha}{\left(\alpha + 2.0\right) + \beta}\right)}^{\left(1 + 3\right)}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999847547237

    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 flip--57.9

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

    7. Taylor expanded around inf 11.0

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{12.0 \cdot \frac{1}{{\alpha}^{2}} - \left(32.0 \cdot \frac{1}{{\alpha}^{3}} + 4.0 \cdot \frac{1}{\alpha}\right)}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}}{2.0}\]

    8. Applied simplify11.0

      \[\leadsto \color{blue}{\frac{\frac{\beta}{2.0}}{2.0 + \left(\alpha + \beta\right)} - \frac{\frac{12.0 - \frac{32.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{4.0}{\alpha}}{2.0}}{1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}}\]

    if -0.9999999847547237 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied flip--0.1

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

    8. Applied flip3--0.1

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

    9. Applied simplify0.1

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

    10. Applied simplify0.1

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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))