Average Error: 15.9 → 3.2
Time: 2.1m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 0.0:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{(\left(\frac{\beta}{\left(2.0 + \alpha\right) + \beta}\right) \cdot \left(\frac{\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} \cdot \frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0 \cdot 1.0}{\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} + 1.0}\right) + \left((\left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right) \cdot \left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right) + \left(\frac{\beta}{\left(2.0 + \alpha\right) + \beta} \cdot \frac{\beta}{\left(2.0 + \alpha\right) + \beta}\right))_*\right))_*}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 0.0

    1. Initial program 60.6

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

    3. Applied div-sub60.6

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

    4. Applied associate-+l-58.7

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

    5. Taylor expanded around inf 10.6

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

    if 0.0 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    6. Applied flip3--0.7

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

    7. Applied simplify0.7

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

    9. Applied flip--0.7

      \[\leadsto \frac{\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{(\left(\frac{\beta}{\left(2.0 + \alpha\right) + \beta}\right) \cdot \color{blue}{\left(\frac{\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} \cdot \frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0 \cdot 1.0}{\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} + 1.0}\right)} + \left((\left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right) \cdot \left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right) + \left(\frac{\beta}{\left(2.0 + \alpha\right) + \beta} \cdot \frac{\beta}{\left(2.0 + \alpha\right) + \beta}\right))_*\right))_*}}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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))