Average Error: 3.9 → 1.4
Time: 9.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)} \le 0.24999999999977907:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}}{\beta + \left(\alpha + 2\right)} \cdot (\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*\\ \end{array}\]

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))) < 0.24999999999977907

    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 associate-+l+0.2

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

    if 0.24999999999977907 < (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1)))

    1. Initial program 7.8

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

    3. Applied associate-+l+7.8

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

    5. Applied *-un-lft-identity7.8

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

    6. Applied div-inv7.8

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

    7. Applied times-frac7.8

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

    8. Simplified7.8

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

    9. Simplified7.8

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

    10. Taylor expanded around 0 2.7

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

    11. Simplified2.7

      \[\leadsto \color{blue}{(\left(\alpha + \beta\right) \cdot 0.25 + 0.5)_*} \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + \left(1.0 + 2\right)}}{\left(\alpha + 2\right) + \beta}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)} \le 0.24999999999977907:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}}{\beta + \left(\alpha + 2\right)} \cdot (\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*\\ \end{array}\]

Runtime

Time bar (total: 9.3m)Debug logProfile

herbie shell --seed 2018274 +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)))