Average Error: 15.9 → 3.5
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - (\left(\frac{\frac{1}{\alpha}}{\alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0} \le 3.6859723283436593 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} - {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {1.0}^{3}\right)}}{(\left(1.0 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)}\right) \cdot 1.0 + \left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)}\right))_*}}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - (\left(\frac{\frac{1}{\alpha}}{\alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0} \le 1.1415152318012226 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - (\left(\frac{\frac{1}{\alpha}}{\alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\left(\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \left(\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (/ beta (+ 2.0 (+ alpha beta))) (fma (/ (/ 1 alpha) alpha) (- 4.0 (/ 8.0 alpha)) (- (/ 2.0 alpha)))) 2.0) < 3.6859723283436593e-295

    1. Initial program 1.6

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

    3. Applied flip3-+1.6

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

    4. Applied simplify1.6

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

    6. Applied flip3-+1.6

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

    if 3.6859723283436593e-295 < (/ (- (/ beta (+ 2.0 (+ alpha beta))) (fma (/ (/ 1 alpha) alpha) (- 4.0 (/ 8.0 alpha)) (- (/ 2.0 alpha)))) 2.0) < 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

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

    if 1.1415152318012226e-05 < (/ (- (/ beta (+ 2.0 (+ alpha beta))) (fma (/ (/ 1 alpha) alpha) (- 4.0 (/ 8.0 alpha)) (- (/ 2.0 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-sqr-sqrt0.1

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

    7. Applied fma-neg0.1

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

Runtime

Time bar (total: 1.8m)Debug logProfile

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