Average Error: 52.3 → 37.8
Time: 2.5m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.883771736890042 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(\left(i + \alpha\right) \cdot \left(i + \beta\right)\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 8.883771736890042e+135

    1. Initial program 50.0

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

    2. Initial simplification50.0

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

    4. Applied *-un-lft-identity50.0

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

    5. Applied times-frac35.0

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

    6. Simplified35.0

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

    7. Simplified35.0

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

    if 8.883771736890042e+135 < beta

    1. Initial program 62.3

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

    2. Initial simplification62.3

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

    4. Applied *-un-lft-identity62.3

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

    5. Applied times-frac55.3

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

    6. Simplified55.3

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

    7. Simplified55.3

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

    8. Taylor expanded around -inf 50.1

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

  4. Final simplification37.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.883771736890042 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(\left(i + \alpha\right) \cdot \left(i + \beta\right)\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes38.737.836.22.438.3%
herbie shell --seed 2018290 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))