Average Error: 52.2 → 38.5
Time: 4.2m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.600225331198575 \cdot 10^{+31} \lor \neg \left(\alpha \le 3.988532239714125 \cdot 10^{+147}\right):\\ \;\;\;\;\left(\left(\left(i + \alpha\right) + \beta\right) \cdot \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\frac{1}{2} \cdot i + \left(\alpha + \beta\right) \cdot \frac{1}{4}}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}}\\ \end{array}\]

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 < 7.600225331198575e+31 or 3.988532239714125e+147 < alpha

    1. Initial program 52.1

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

    3. Applied *-un-lft-identity52.1

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

    4. Applied times-frac39.4

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

    5. Applied times-frac39.5

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

    6. Simplified39.5

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

    7. Taylor expanded around 0 38.8

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

    8. Simplified38.8

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

    if 7.600225331198575e+31 < alpha < 3.988532239714125e+147

    1. Initial program 53.4

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

    3. Applied add-sqr-sqrt53.4

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

    4. Applied times-frac35.9

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

    5. Applied times-frac35.9

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

  4. Final simplification38.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.600225331198575 \cdot 10^{+31} \lor \neg \left(\alpha \le 3.988532239714125 \cdot 10^{+147}\right):\\ \;\;\;\;\left(\left(\left(i + \alpha\right) + \beta\right) \cdot \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}\right) \cdot \frac{\frac{1}{2} \cdot i + \left(\alpha + \beta\right) \cdot \frac{1}{4}}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right) - 1.0}}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes40.238.536.34.044.3%
herbie shell --seed 2018263 
(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)))