Average Error: 52.6 → 37.9
Time: 4.6m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.8202630269532325 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\left(i + \beta\right) \cdot \left(i + \alpha\right)}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\left(i + \alpha\right) + \beta}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) + \beta}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(i \cdot \frac{1}{4}\right)\\ \end{array}\]

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 alpha < 3.8202630269532325e+143

    1. Initial program 50.5

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

    2. Initial simplification44.5

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

    4. Applied times-frac35.3

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

    5. Simplified35.3

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

    if 3.8202630269532325e+143 < alpha

    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 simplification59.0

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

    3. Taylor expanded around 0 50.2

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

  4. Final simplification37.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.8202630269532325 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\left(i + \beta\right) \cdot \left(i + \alpha\right)}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\left(i + \alpha\right) + \beta}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) + \beta}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(i \cdot \frac{1}{4}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed 2018219 
(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)))