Average Error: 15.9 → 3.5
Time: 1.2m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0}}{\beta + \left(2.0 + \alpha\right)} - \left(\frac{4.0 - \frac{8.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{1}{\alpha}}{1}\right) \le 3.685972328343673 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + {1.0}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0}}{\beta + \left(2.0 + \alpha\right)} - \left(\frac{4.0 - \frac{8.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{1}{\alpha}}{1}\right) \le 1.1415152318012226 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\beta}{2.0}}{\beta + \left(2.0 + \alpha\right)} - \left(\frac{4.0 - \frac{8.0}{\alpha}}{2.0 \cdot \left(\alpha \cdot \alpha\right)} - \frac{\frac{1}{\alpha}}{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + {1.0}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (/ beta 2.0) (+ beta (+ 2.0 alpha))) (- (/ (- 4.0 (/ 8.0 alpha)) (* 2.0 (* alpha alpha))) (/ (/ 1 alpha) 1))) < 3.685972328343673e-295 or 1.1415152318012226e-05 < (- (/ (/ beta 2.0) (+ beta (+ 2.0 alpha))) (- (/ (- 4.0 (/ 8.0 alpha)) (* 2.0 (* alpha alpha))) (/ (/ 1 alpha) 1)))

    1. Initial program 0.8

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

    3. Applied flip3-+0.8

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

    if 3.685972328343673e-295 < (- (/ (/ beta 2.0) (+ beta (+ 2.0 alpha))) (- (/ (- 4.0 (/ 8.0 alpha)) (* 2.0 (* alpha alpha))) (/ (/ 1 alpha) 1))) < 1.1415152318012226e-05

    1. Initial program 59.4

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

    3. Applied div-sub59.4

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

    4. Applied associate-+l-57.5

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

    5. Taylor expanded around inf 11.2

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

    6. Applied simplify11.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))