Average Error: 3.5 → 2.2
Time: 4.5m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 5.5279191982440455 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 2} \cdot \frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 5.5279191982440455e+171

    1. Initial program 1.3

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

    3. Applied div-inv1.3

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

    if 5.5279191982440455e+171 < beta

    1. Initial program 15.6

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

    2. Taylor expanded around inf 6.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.5279191982440455 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 2} \cdot \frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2018348 
(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)))