Average Error: 52.9 → 37.5
Time: 4.3m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.9186205165128145 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\left(i + \alpha\right) \cdot \left(i + \beta\right)}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\beta + \left(i + \alpha\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\beta + \left(i + \alpha\right)} \cdot \frac{\sqrt{\beta + \left(i + \alpha\right)}}{\beta \cdot \left(i \cdot 4 + \left(2 \cdot \alpha + \beta\right)\right) - 1.0}\right) \cdot \left(i \cdot \frac{1}{4}\right)\\ \end{array}\]

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 < 4.9186205165128145e+153

    1. Initial program 50.9

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

    2. Initial simplification44.7

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

    4. Applied times-frac35.2

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

    5. Simplified35.2

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

    if 4.9186205165128145e+153 < beta

    1. Initial program 62.5

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

    2. Initial simplification58.6

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

    3. Taylor expanded around 0 48.7

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot i\right)} \cdot \frac{\left(\alpha + i\right) + \beta}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \frac{\left(\alpha + i\right) + \beta}{\color{blue}{1 \cdot \left(\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0\right)}}\]

    6. Applied add-sqr-sqrt48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \frac{\color{blue}{\sqrt{\left(\alpha + i\right) + \beta} \cdot \sqrt{\left(\alpha + i\right) + \beta}}}{1 \cdot \left(\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0\right)}\]

    7. Applied times-frac48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \color{blue}{\left(\frac{\sqrt{\left(\alpha + i\right) + \beta}}{1} \cdot \frac{\sqrt{\left(\alpha + i\right) + \beta}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}\right)}\]

    8. Simplified48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \left(\color{blue}{\sqrt{\beta + \left(i + \alpha\right)}} \cdot \frac{\sqrt{\left(\alpha + i\right) + \beta}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}\right)\]

    9. Taylor expanded around inf 48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \left(\sqrt{\beta + \left(i + \alpha\right)} \cdot \frac{\sqrt{\left(\alpha + i\right) + \beta}}{\color{blue}{\left(2 \cdot \left(\beta \cdot \alpha\right) + \left({\beta}^{2} + 4 \cdot \left(i \cdot \beta\right)\right)\right)} - 1.0}\right)\]

    10. Simplified48.7

      \[\leadsto \left(\frac{1}{4} \cdot i\right) \cdot \left(\sqrt{\beta + \left(i + \alpha\right)} \cdot \frac{\sqrt{\left(\alpha + i\right) + \beta}}{\color{blue}{\beta \cdot \left(\left(2 \cdot \alpha + \beta\right) + i \cdot 4\right)} - 1.0}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification37.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.9186205165128145 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\left(i + \alpha\right) \cdot \left(i + \beta\right)}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\beta + \left(i + \alpha\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\beta + \left(i + \alpha\right)} \cdot \frac{\sqrt{\beta + \left(i + \alpha\right)}}{\beta \cdot \left(i \cdot 4 + \left(2 \cdot \alpha + \beta\right)\right) - 1.0}\right) \cdot \left(i \cdot \frac{1}{4}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018220 
(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)))