Average Error: 15.8 → 16.0
Time: 3.1m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\alpha \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-1.0\right))_*}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 15.8

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

  3. Applied div-sub15.8

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

  4. Applied associate-+l-15.3

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

  6. Applied div-inv15.4

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

  7. Applied fma-neg16.0

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))