Average Error: 23.5 → 12.0
Time: 6.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.120200533334092 \cdot 10^{+77}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\beta + \alpha\right) + 1.0\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 3.083993294069994 \cdot 10^{+157} \lor \neg \left(\alpha \le 4.686585818956074 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}} \cdot \frac{1}{\sqrt{\sqrt{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}} \cdot \left|\sqrt[3]{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}\right|}\right) \cdot \left(\beta + \alpha\right)}{2.0}\\ \end{array}\]

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 3 regimes

  2. if alpha < 8.120200533334092e+77

    1. Initial program 12.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity12.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity12.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac2.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac2.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified2.2

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt2.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    10. Applied associate-/r*2.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
    11. Using strategy rm

    12. Applied add-exp-log2.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]

    if 8.120200533334092e+77 < alpha < 3.083993294069994e+157 or 4.686585818956074e+231 < alpha

    1. Initial program 53.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around -inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified41.3

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}}{2.0}\]

    if 3.083993294069994e+157 < alpha < 4.686585818956074e+231

    1. Initial program 63.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity63.2

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity63.2

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac43.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac42.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified42.9

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt43.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    10. Applied associate-/r*43.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt43.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}} + 1.0}{2.0}\]

    13. Applied sqrt-prod43.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\color{blue}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}} + 1.0}{2.0}\]

    14. Applied add-sqr-sqrt43.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    15. Applied *-un-lft-identity43.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{1 \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    16. Applied times-frac43.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    17. Applied times-frac43.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right)} + 1.0}{2.0}\]

    18. Simplified43.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \left(\color{blue}{\frac{1}{\sqrt{\sqrt{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}} \cdot \left|\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}\right|}} \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right) + 1.0}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.120200533334092 \cdot 10^{+77}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\beta + \alpha\right) + 1.0\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 3.083993294069994 \cdot 10^{+157} \lor \neg \left(\alpha \le 4.686585818956074 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}} \cdot \frac{1}{\sqrt{\sqrt{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}} \cdot \left|\sqrt[3]{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}\right|}\right) \cdot \left(\beta + \alpha\right)}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 6.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.312.05.96.44.4%
herbie shell --seed 2018295 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))