Average Error: 23.7 → 12.0
Time: 7.4m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.2668330762841353 \cdot 10^{+95}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right) \cdot \left(\left(\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{1}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 9.840915374610538 \cdot 10^{+161} \lor \neg \left(\alpha \le 2.7326470423674704 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\frac{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}} + 1.0}{2.0}\\ \end{array}\]

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 < 1.2668330762841353e+95

    1. Initial program 12.9

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

    3. Applied *-un-lft-identity12.9

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

    4. Applied times-frac2.8

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

    5. Applied associate-/l*2.8

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

    6. Simplified2.8

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

    8. Applied add-cbrt-cube2.8

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

    10. Applied div-inv2.8

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

    if 1.2668330762841353e+95 < alpha < 9.840915374610538e+161 or 2.7326470423674704e+203 < alpha

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 41.1

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

    3. Simplified41.1

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

    if 9.840915374610538e+161 < alpha < 2.7326470423674704e+203

    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

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

    4. Applied times-frac42.0

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

    5. Applied associate-/l*42.0

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

    6. Simplified42.0

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

    8. Applied add-sqr-sqrt42.3

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

    9. Applied associate-/l*42.3

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.2668330762841353 \cdot 10^{+95}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right) \cdot \left(\left(\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{1}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 9.840915374610538 \cdot 10^{+161} \lor \neg \left(\alpha \le 2.7326470423674704 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\frac{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}} + 1.0}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 7.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.512.06.06.57.6%
herbie shell --seed 2018351 
(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))