Average Error: 15.8 → 2.9
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 4.2125482990762464 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \end{array}\]

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) < 4.2125482990762464e-08

    1. Initial program 59.8

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

    3. Applied div-sub59.7

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

    4. Applied associate-+l-57.9

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

    5. Taylor expanded around inf 10.8

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

    if 4.2125482990762464e-08 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 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

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))