Average Error: 52.5 → 37.9
Time: 3.1m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.6893555258837206 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}{\left(\left(2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) - 1.0}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}{\left(\left(2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 alpha < 2.6893555258837206e+131

    1. Initial program 50.1

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

    3. Applied associate-/l*34.9

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

    5. Applied distribute-lft-in34.9

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

    7. Applied add-sqr-sqrt34.9

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

    if 2.6893555258837206e+131 < alpha

    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. Using strategy rm

    3. Applied associate-/l*54.4

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

    5. Applied distribute-lft-in54.4

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

    6. Taylor expanded around inf 50.2

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

  4. Final simplification37.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.6893555258837206 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}{\left(\left(2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) - 1.0}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}{\left(\left(2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes38.737.936.32.433.9%
herbie shell --seed 2018340 
(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)))