Average Error: 24.1 → 4.2
Time: 3.8m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\beta + \alpha} \cdot \left(\sqrt{\beta + \alpha} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0}{2.0} \le 5.5065840776080677 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\left(\frac{2.0}{\alpha} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) - \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot 2\right) + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (sqrt (+ beta alpha)) (* (sqrt (+ beta alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0) < 5.5065840776080677e-11

    1. Initial program 62.2

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

    3. Applied *-un-lft-identity62.2

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

    4. Applied add-cube-cbrt62.0

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

    5. Applied times-frac60.4

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

    6. Applied times-frac60.3

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

    7. Applied simplify60.3

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

    9. Applied add-log-exp60.3

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

    10. Taylor expanded around inf 21.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2.0 \cdot \frac{1}{\alpha}\right) - \left(6.0 \cdot \frac{\beta}{{\alpha}^{2}} + \left(2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}} + 4.0 \cdot \frac{1}{{\alpha}^{2}}\right)\right)}}{2.0}\]

    11. Applied simplify19.2

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{2.0}{\alpha} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) - \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot 2\right) + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6.0}{\alpha}\right)}{2.0}}\]

    if 5.5065840776080677e-11 < (/ (+ (* (sqrt (+ beta alpha)) (* (sqrt (+ beta alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0)

    1. Initial program 14.0

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

    3. Applied *-un-lft-identity14.0

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

    4. Applied *-un-lft-identity14.0

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

    5. Applied times-frac0.2

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

    6. Applied times-frac0.2

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

    7. Applied simplify0.2

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(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))