Average Error: 23.7 → 11.2
Time: 4.7m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.4166996658578758 \cdot 10^{+164}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{\sqrt{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}}{\sqrt{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

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 2 regimes

  2. if alpha < 1.4166996658578758e+164

    1. Initial program 16.4

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

    3. Applied associate-/l*5.7

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

    5. Applied add-sqr-sqrt5.7

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

    6. Applied associate-/r*5.7

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

    if 1.4166996658578758e+164 < alpha

    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 associate-/l*48.9

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

    5. Applied add-cube-cbrt49.0

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

    6. Applied *-un-lft-identity49.0

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

    7. Applied times-frac49.0

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

    8. Applied associate-/l*49.0

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

    9. Taylor expanded around inf 41.1

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

    10. Simplified41.1

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.4166996658578758 \cdot 10^{+164}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{\sqrt{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}}{\sqrt{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018346 
(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))