Average Error: 3.3 → 1.4
Time: 5.7m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)} \le 0.08333332649330999:\\ \;\;\;\;\frac{\frac{\left(\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\beta + \alpha\right) + 0.5}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

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)) < 0.08333332649330999

    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 div-inv0.2

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

    4. Applied simplify0.2

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

    if 0.08333332649330999 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 6.5

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

    2. Taylor expanded around 0 6.5

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

    3. Applied simplify2.6

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

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)} \le 0.08333332649330999:\\ \;\;\;\;\frac{\frac{\left(\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\beta + \alpha\right) + 0.5}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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)))