Average Error: 3.6 → 2.8
Time: 4.8m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.3806542021457376 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + \left(1.0 + 2\right)}}{\left(2 + \alpha\right) + \beta} \cdot e^{\log \left(\left(\beta \cdot \alpha + \beta\right) + \left(\alpha + 1.0\right)\right) - \log \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.3806542021457376e+161

    1. Initial program 1.2

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

    2. Applied simplify8.0

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

    4. Applied *-un-lft-identity8.0

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

    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

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

    9. Applied add-exp-log2.9

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

    10. Applied add-exp-log1.8

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

    11. Applied div-exp1.8

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

    if 1.3806542021457376e+161 < alpha

    1. Initial program 16.2

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

    2. Applied simplify18.0

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

    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: 4.8m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))