Average Error: 16.2 → 3.1
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le -1.621792723509092 \cdot 10^{-307}:\\ \;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le 9.069375325087562 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < -1.621792723509092e-307

    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

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

    if -1.621792723509092e-307 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 9.069375325087562e-06

    1. Initial program 59.1

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

    3. Applied div-sub59.1

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

    4. Applied associate-+l-57.3

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

    5. Taylor expanded around inf 11.3

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

    if 9.069375325087562e-06 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 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 add-log-exp0.1

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +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))