Error

Derivation

1. Initial program 23.7

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

2. Initial simplification12.3

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

4. Applied expm1-log1p-u12.3

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

6. Applied add-exp-log12.3

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

8. Applied add-sqr-sqrt12.5

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

9. Applied log-prod12.3

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

10. Simplified12.3

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

11. Final simplification12.3

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

Runtime

Time bar (total: 39.9s)Debug logProfile

herbie shell --seed 2018221 +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))