Average Error: 23.3 → 12.6
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\frac{{\left({\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right)}^{3}\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right)}^{3} \cdot {\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} - {1.0}^{3} \cdot {\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right)}^{3}\right)}}{\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + \left(1.0 \cdot 1.0 - 1.0 \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right)\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.3

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

  2. Initial simplification12.6

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

  4. Applied fma-udef12.6

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

  6. Applied flip3-+12.6

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

  8. Applied flip3-+12.6

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

  9. Final simplification12.6

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018249 +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))