Average Error: 23.7 → 12.3
Time: 2.5m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.887135031661262 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \mathbf{elif}\;\alpha \le 1.0066251309182657 \cdot 10^{+130} \lor \neg \left(\alpha \le 6.736335860516646 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1.0}^{3} + \log \left(e^{{\left(\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)}^{3}}\right)}{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + \left(1.0 \cdot 1.0 - \frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot 1.0\right)}}{2.0}\\ \end{array}\]

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 < 5.887135031661262e+62

    1. Initial program 12.0

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

    3. Applied *-un-lft-identity12.0

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

    4. Applied times-frac1.6

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

    5. Simplified1.6

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

    7. Applied *-un-lft-identity1.6

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

    8. Applied times-frac1.6

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

    9. Simplified1.6

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

    if 5.887135031661262e+62 < alpha < 1.0066251309182657e+130 or 6.736335860516646e+165 < alpha

    1. Initial program 55.2

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

    2. Taylor expanded around inf 41.7

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

    3. Simplified41.7

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

    if 1.0066251309182657e+130 < alpha < 6.736335860516646e+165

    1. Initial program 50.8

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

    3. Applied *-un-lft-identity50.8

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

    4. Applied times-frac34.1

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

    5. Simplified34.1

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

    7. Applied flip3-+34.1

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

    9. Applied add-log-exp34.1

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.887135031661262 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \mathbf{elif}\;\alpha \le 1.0066251309182657 \cdot 10^{+130} \lor \neg \left(\alpha \le 6.736335860516646 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1.0}^{3} + \log \left(e^{{\left(\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)}^{3}}\right)}{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + \left(1.0 \cdot 1.0 - \frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot 1.0\right)}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018278 
(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))