Average Error: 52.1 → 37.6
Time: 5.6m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.871096467066142 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}} \cdot \frac{i \cdot \left(\beta + \alpha\right) + \left(\alpha \cdot \beta + i \cdot i\right)}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\frac{i}{\left(\beta + \alpha\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 < 1.871096467066142e+140

    1. Initial program 49.9

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

    2. Initial simplification36.2

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

    4. Applied add-sqr-sqrt36.2

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

    5. Applied times-frac36.2

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

    6. Applied *-un-lft-identity36.2

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

    7. Applied times-frac35.0

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

    8. Simplified35.0

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

    if 1.871096467066142e+140 < beta

    1. Initial program 62.4

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

    2. Initial simplification56.8

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

    3. Taylor expanded around inf 49.8

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

  4. Final simplification37.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.871096467066142 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}} \cdot \frac{i \cdot \left(\beta + \alpha\right) + \left(\alpha \cdot \beta + i \cdot i\right)}{\frac{\sqrt{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}}{\frac{i}{\left(\beta + \alpha\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 5.6m)Debug logProfile

herbie shell --seed 2018221 
(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)))