Average Error: 23.9 → 13.2
Time: 55.3s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left(\log \left(e^{\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{1}{\frac{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}{\beta - \alpha}} + 1.0}\right)\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

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

  2. Initial simplification13.2

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

  4. Applied *-un-lft-identity13.2

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

  5. Applied associate-/l*13.2

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

  7. Applied add-log-exp13.2

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

  9. Applied add-exp-log13.2

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

  11. Applied fma-udef13.2

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

  12. Final simplification13.2

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

Runtime

Time bar (total: 55.3s)Debug logProfile

herbie shell --seed 2018255 +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))