Average Error: 3.6 → 2.5
Time: 7.4m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 5.105635182788669 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + \left(1.0 + 2\right)}}{\left(2 + \alpha\right) + \beta} \cdot \frac{\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)}{\left(2 + \alpha\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 5.105635182788669e+161

    1. Initial program 1.0

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

    2. Applied simplify8.1

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

    4. Applied *-un-lft-identity8.1

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

    5. Applied times-frac1.9

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

    7. Applied associate-/r*1.5

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

    if 5.105635182788669e+161 < beta

    1. Initial program 17.3

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

    2. Applied simplify18.4

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

    3. Taylor expanded around inf 8.0

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

Runtime

Time bar (total: 7.4m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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)))