Average Error: 3.7 → 1.7
Time: 2.9m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 0.08333331283332793:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{\left(\alpha + 1.0\right) + \left(\beta \cdot \alpha + \beta\right)}{\left(\beta + 2\right) + \alpha}} \cdot \sqrt{\frac{\left(\alpha + 1.0\right) + \left(\beta \cdot \alpha + \beta\right)}{\left(\beta + 2\right) + \alpha}}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 0.08333331283332793

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied simplify0.2

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

    5. Applied simplify0.2

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

    if 0.08333331283332793 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 7.4

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

    2. Taylor expanded around 0 7.4

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

    3. Applied simplify3.2

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

  4. Applied simplify1.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 0.08333331283332793:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{\left(\alpha + 1.0\right) + \left(\beta \cdot \alpha + \beta\right)}{\left(\beta + 2\right) + \alpha}} \cdot \sqrt{\frac{\left(\alpha + 1.0\right) + \left(\beta \cdot \alpha + \beta\right)}{\left(\beta + 2\right) + \alpha}}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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)))