Average Error: 16.2 → 16.5
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{(\left(\beta - \alpha\right) \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0)_*}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 16.2

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

  3. Applied div-inv16.3

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

  4. Applied fma-def16.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018195 +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))