Average Error: 3.8 → 2.3
Time: 9.2m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\alpha + 2\right) + \left(\beta + 1.0\right)\right)} \le 6.91459205818515 \cdot 10^{-310}:\\ \;\;\;\;\frac{0.5 + 0.25 \cdot \left(\beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\alpha + 2\right) + \left(\beta + 1.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + \left(\beta \cdot \alpha + 1.0\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + \left(2 + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array}\]

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 (/ (+ 0.5 (* 0.25 (+ beta alpha))) (* (+ (+ beta alpha) 2) (+ (+ alpha 2) (+ beta 1.0)))) < 6.91459205818515e-310

    1. Initial program 12.0

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

    2. Taylor expanded around 0 59.3

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

    3. Applied simplify7.2

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

    if 6.91459205818515e-310 < (/ (+ 0.5 (* 0.25 (+ beta alpha))) (* (+ (+ beta alpha) 2) (+ (+ alpha 2) (+ beta 1.0))))

    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. Applied simplify0.2

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

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed 2020178 
(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)))