Average Error: 3.5 → 1.6
Time: 5.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta + \alpha \le 1.8674321140604034 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{\beta \cdot \alpha + \left(\alpha + \left(\beta + 1.0\right)\right)}}{\left(\beta + \alpha\right) + 2} \cdot \frac{\sqrt[3]{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1.0}}{1}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (+ alpha beta) < 1.8674321140604034e+196

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied add-cube-cbrt0.9

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

    5. Applied times-frac0.9

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

    6. Applied simplify0.9

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

    if 1.8674321140604034e+196 < (+ alpha beta)

    1. Initial program 13.7

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

    2. Taylor expanded around 0 58.9

      \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    3. Applied simplify4.2

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\alpha + \beta\right) + 0.5}{\left(\left(2 + 1.0\right) + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\beta + \alpha \le 1.8674321140604034 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{\beta \cdot \alpha + \left(\alpha + \left(\beta + 1.0\right)\right)}}{\left(\beta + \alpha\right) + 2} \cdot \frac{\sqrt[3]{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1.0}}{1}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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)))