Average Error: 3.4 → 1.5
Time: 8.9m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 3.6275127866633337 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{\left(\beta + \alpha\right) + (\alpha \cdot \beta + 1.0)_*} \cdot \sqrt[3]{\left(\beta + \alpha\right) + (\alpha \cdot \beta + 1.0)_*}\right) \cdot \frac{\sqrt[3]{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 3.6275127866633337e+96

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-cube-cbrt0.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    5. Applied times-frac0.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{1} \cdot \frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied simplify0.4

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{(\alpha \cdot \beta + 1.0)_* + \left(\beta + \alpha\right)} \cdot \sqrt[3]{(\alpha \cdot \beta + 1.0)_* + \left(\beta + \alpha\right)}\right)} \cdot \frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Applied simplify0.4

      \[\leadsto \frac{\frac{\left(\sqrt[3]{(\alpha \cdot \beta + 1.0)_* + \left(\beta + \alpha\right)} \cdot \sqrt[3]{(\alpha \cdot \beta + 1.0)_* + \left(\beta + \alpha\right)}\right) \cdot \color{blue}{\frac{\sqrt[3]{\left(\beta + 1.0\right) + (\beta \cdot \alpha + \alpha)_*}}{2 + \left(\beta + \alpha\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    if 3.6275127866633337e+96 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 63.0

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

    2. Taylor expanded around inf 20.9

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + 2.0 \cdot \frac{1}{{\alpha}^{2}}\right) - 1.0 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    3. Applied simplify20.9

      \[\leadsto \color{blue}{\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 3.6275127866633337 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{\left(\beta + \alpha\right) + (\alpha \cdot \beta + 1.0)_*} \cdot \sqrt[3]{\left(\beta + \alpha\right) + (\alpha \cdot \beta + 1.0)_*}\right) \cdot \frac{\sqrt[3]{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 8.9m)Debug logProfile

herbie shell --seed 2018206 +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)))