Average Error: 3.5 → 2.1
Time: 7.1m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.908868903096529 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(1.0 + \beta\right) + (\beta \cdot \alpha + \alpha)_*}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt{\frac{\frac{\left(1.0 + \beta\right) + (\beta \cdot \alpha + \alpha)_*}{\left(\beta + 2\right) + \alpha}}{\left(\beta + 2\right) + \alpha}}}{\left(\alpha + 2\right) + \left(1.0 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{2 + \left(\alpha + \beta\right)}}}{(\left((\left(\sqrt{\frac{1}{8}}\right) \cdot \alpha + \left(\frac{\beta}{\sqrt{8}}\right))_*\right) \cdot 6.0 + \left((\left(\sqrt{8} \cdot 1.0\right) \cdot \left(-\left(\alpha + \beta\right)\right) + \left(\sqrt{8} \cdot 1.0\right))_*\right))_*}}{1.0 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.908868903096529e+164

    1. Initial program 1.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied log1p-expm1-u1.2

      \[\leadsto \frac{\color{blue}{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity1.2

      \[\leadsto \frac{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}}\]

    6. Applied add-sqr-sqrt1.3

      \[\leadsto \frac{\color{blue}{\sqrt{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)} \cdot \sqrt{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]

    7. Applied times-frac1.3

      \[\leadsto \color{blue}{\frac{\sqrt{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}}{1} \cdot \frac{\sqrt{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}}\]

    8. Simplified1.3

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}} \cdot \frac{\sqrt{\log_* (1 + (e^{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} - 1)^*)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    9. Simplified1.3

      \[\leadsto \sqrt{\frac{\frac{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}} \cdot \color{blue}{\frac{\sqrt{\frac{\frac{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}{\left(2 + \beta\right) + \alpha}}{\left(2 + \beta\right) + \alpha}}}{\left(\beta + 1.0\right) + \left(\alpha + 2\right)}}\]

    if 4.908868903096529e+164 < beta

    1. Initial program 16.8

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt16.8

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    4. Applied *-un-lft-identity16.8

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    5. Applied times-frac16.9

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied associate-/l*16.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Simplified16.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    8. Taylor expanded around 0 7.3

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\color{blue}{\left(6.0 \cdot \frac{\beta}{\sqrt{8}} + \left(6.0 \cdot \left(\sqrt{\frac{1}{8}} \cdot \alpha\right) + 1.0 \cdot \sqrt{8}\right)\right) - \left(1.0 \cdot \left(\sqrt{8} \cdot \alpha\right) + 1.0 \cdot \left(\sqrt{8} \cdot \beta\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    9. Simplified7.1

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\color{blue}{(\left((\left(\sqrt{\frac{1}{8}}\right) \cdot \alpha + \left(\frac{\beta}{\sqrt{8}}\right))_*\right) \cdot 6.0 + \left((\left(\sqrt{8} \cdot 1.0\right) \cdot \left(-\left(\beta + \alpha\right)\right) + \left(\sqrt{8} \cdot 1.0\right))_*\right))_*}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.908868903096529 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(1.0 + \beta\right) + (\beta \cdot \alpha + \alpha)_*}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt{\frac{\frac{\left(1.0 + \beta\right) + (\beta \cdot \alpha + \alpha)_*}{\left(\beta + 2\right) + \alpha}}{\left(\beta + 2\right) + \alpha}}}{\left(\alpha + 2\right) + \left(1.0 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{2 + \left(\alpha + \beta\right)}}}{(\left((\left(\sqrt{\frac{1}{8}}\right) \cdot \alpha + \left(\frac{\beta}{\sqrt{8}}\right))_*\right) \cdot 6.0 + \left((\left(\sqrt{8} \cdot 1.0\right) \cdot \left(-\left(\alpha + \beta\right)\right) + \left(\sqrt{8} \cdot 1.0\right))_*\right))_*}}{1.0 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))