Average Error: 15.9 → 3.2
Time: 3.8m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0} \le 1.360984718392274 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} + \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0 + 1.0 \cdot 1.0\right)\right) - \left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left(2.0 + \left(\alpha + \beta\right)\right)}{\left(2.0 + \left(\alpha + \beta\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0 + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0} \le 3.2412261942033065 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - (\alpha \cdot \left(\frac{1}{2.0 + \left(\alpha + \beta\right)}\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 1.360984718392274e-305

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied flip3--0.4

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

    7. Applied frac-sub0.5

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

    8. Taylor expanded around inf 0.8

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\color{blue}{0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}{2.0}\]

    if 1.360984718392274e-305 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 3.2412261942033065e-10

    1. Initial program 60.2

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

    3. Applied div-sub60.2

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

    4. Applied associate-+l-58.3

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

    5. Taylor expanded around inf 11.0

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

    if 3.2412261942033065e-10 < (/ (- (/ 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 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied div-inv0.3

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

    7. Applied fma-neg0.3

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

  4. Applied simplify3.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0} \le 1.360984718392274 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} + \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0 + 1.0 \cdot 1.0\right)\right) - \left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left(2.0 + \left(\alpha + \beta\right)\right)}{\left(2.0 + \left(\alpha + \beta\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0 + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0} \le 3.2412261942033065 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(2.0 \cdot \frac{1}{\alpha} + \frac{1}{{\alpha}^{3}} \cdot 8.0\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - (\alpha \cdot \left(\frac{1}{2.0 + \left(\alpha + \beta\right)}\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018167 +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))