Average Error: 3.7 → 1.7
Time: 8.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \le 4.395216163716034 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{\left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right) + 1.0}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + 2\right) + \left(\beta + \alpha\right)} \cdot \frac{1}{\left(2 + \alpha\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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))) < 4.395216163716034e+107

    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. Applied simplify6.7

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

    4. Applied *-un-lft-identity6.7

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

    5. Applied times-frac1.0

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

    7. Applied div-inv1.0

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

    8. Applied associate-*r*1.1

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

    9. Applied simplify1.0

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

    11. Applied associate-/r*0.2

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

    if 4.395216163716034e+107 < (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1)))

    1. Initial program 57.8

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

    2. Applied simplify62.8

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

    3. Taylor expanded around inf 24.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 8.3m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))