Error

Derivation

1. Split input into 2 regimes

2. if (/ (fma (/ (+ beta alpha) (+ (fma i 2 beta) (+ 2.0 alpha))) (* (- beta alpha) (* (sqrt (/ 1 (fma 2 i (+ beta alpha)))) (sqrt (/ 1 (fma 2 i (+ beta alpha)))))) 1.0) 2.0) < 6.603606550470431e-13

   1. Initial program 62.5

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

   2. Applied simplify60.4

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

   4. Applied add-exp-log60.4

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

   6. Applied add-cbrt-cube60.4

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

   7. Applied simplify60.4

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

   8. Taylor expanded around inf 33.3

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

   9. Applied simplify30.0

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

   if 6.603606550470431e-13 < (/ (fma (/ (+ beta alpha) (+ (fma i 2 beta) (+ 2.0 alpha))) (* (- beta alpha) (* (sqrt (/ 1 (fma 2 i (+ beta alpha)))) (sqrt (/ 1 (fma 2 i (+ beta alpha)))))) 1.0) 2.0)

   1. Initial program 13.5

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

   2. Applied simplify0.3

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

   4. Applied add-cube-cbrt0.5

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

   5. Applied *-un-lft-identity0.5

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

   6. Applied times-frac0.5

      \[\leadsto \frac{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*}} \cdot \frac{\beta - \alpha}{\sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*}}\right)} + 1.0)_*}{2.0}\]
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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))