Average Error: 24.1 → 11.8
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.192322497084979 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right) \cdot \left(\left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right) \cdot \left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha} + 2.0}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.192322497084979e+138

    1. Initial program 15.1

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

    3. Applied *-un-lft-identity15.1

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

    4. Applied times-frac4.7

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

    5. Simplified4.7

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

    7. Applied div-inv4.7

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

    9. Applied add-cbrt-cube4.7

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

    if 3.192322497084979e+138 < alpha

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 41.7

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

    3. Simplified41.7

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.192322497084979 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right) \cdot \left(\left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right) \cdot \left(\frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta - \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha} + 2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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))