Average Error: 16.0 → 3.6
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le 1.583875705995391 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\frac{{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} \cdot {1.0}^{3}\right)}}{(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) \cdot 1.0 + \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right))_*}}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le 8.499631403808008 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{(e^{\log_* (1 + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3})} - 1)^* - {1.0}^{3}}{(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) \cdot 1.0 + \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right))_*}}{2.0}\\ \end{array}\]

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) < 1.583875705995391e-300

    1. Initial program 1.3

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

    3. Applied div-sub1.3

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

    4. Applied associate-+l-1.3

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

    6. Applied flip3--1.4

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

    7. Applied simplify1.4

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

    9. Applied flip3--1.4

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

    if 1.583875705995391e-300 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 8.499631403808008e-06

    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-57.4

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

    5. Taylor expanded around inf 11.8

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

    if 8.499631403808008e-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 flip3--0.1

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

    7. Applied simplify0.1

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

    9. Applied expm1-log1p-u0.1

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018214 +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))