Average Error: 16.4 → 3.5
Time: 2.5m
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{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 1.8392430812403376 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\frac{{\left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} + {\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} \cdot {1.0}^{3}\right)}}{\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\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 1.5895035384958886 \cdot 10^{-07}:\\ \;\;\;\;\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{\left(\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right) - \frac{2.0 + \left(\alpha + \beta\right)}{\beta} \cdot \left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} - {1.0}^{3}\right)}{\frac{2.0 + \left(\alpha + \beta\right)}{\beta} \cdot \left(\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\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) < 1.8392430812403376e-300

    1. Initial program 1.0

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

    3. Applied div-sub1.0

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

    4. Applied associate-+l-1.0

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

    6. Applied flip3--1.0

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

    8. Applied flip3--1.0

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

    if 1.8392430812403376e-300 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 1.5895035384958886e-07

    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-sub59.8

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

    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 11.6

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

    if 1.5895035384958886e-07 < (/ (- (/ 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 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied clear-num0.2

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    7. Using strategy rm

    8. Applied flip3--0.2

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]

    9. Applied frac-sub0.2

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

  4. Applied simplify3.5

    \[\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 1.8392430812403376 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \frac{\frac{{\left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3}\right)}^{3} - {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} + {\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} \cdot {1.0}^{3}\right)}}{\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\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 1.5895035384958886 \cdot 10^{-07}:\\ \;\;\;\;\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{\left(\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right) - \frac{2.0 + \left(\alpha + \beta\right)}{\beta} \cdot \left({\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}^{3} - {1.0}^{3}\right)}{\frac{2.0 + \left(\alpha + \beta\right)}{\beta} \cdot \left(\left(1.0 \cdot 1.0 + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\alpha}{2.0 + \left(\alpha + \beta\right)}\right)}}{2.0}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))