Average Error: 23.7 → 11.1
Time: 3.7m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.8461140640358156 \cdot 10^{+191}:\\ \;\;\;\;\frac{e^{\log \left((\left(\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}\right) \cdot \left(\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\right) + 1.0)_*\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.8461140640358156e+191

    1. Initial program 18.3

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

    3. Applied associate-/l*6.9

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

    5. Applied *-un-lft-identity6.9

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

    6. Applied associate-/l*6.9

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

    8. Applied div-inv6.9

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

    9. Applied fma-def6.8

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

    11. Applied add-exp-log6.8

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

    if 3.8461140640358156e+191 < 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*50.3

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

    5. Applied *-un-lft-identity50.3

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

    6. Applied associate-/l*50.4

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

    8. Applied div-inv50.3

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

    9. Applied fma-def51.4

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

    10. Taylor expanded around inf 42.6

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

    11. Simplified42.6

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.8461140640358156 \cdot 10^{+191}:\\ \;\;\;\;\frac{e^{\log \left((\left(\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}\right) \cdot \left(\frac{1}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\right) + 1.0)_*\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(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))