\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
Test:
Octave 3.8, oct_fill_randg
Bits:
128 bits
Bits error versus a
Bits error versus rand
Time: 18.4 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\((\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot a - 3.0}}\right) * rand + \left(a - \frac{1.0}{3.0}\right))_*\)
  1. Started with
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
    0.1
  2. Using strategy rm
    0.1
  3. Applied distribute-lft-in to get
    \[\color{red}{\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)} \leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)}\]
    0.1
  4. Applied simplify to get
    \[\color{red}{\left(a - \frac{1.0}{3.0}\right) \cdot 1} + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right)} + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
    0.1
  5. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + \color{red}{\left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)} \leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}}\]
    8.5
  6. Applied taylor to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{9 \cdot a - 3.0}}\]
    8.5
  7. Taylor expanded around 0 to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{\color{red}{9 \cdot a - 3.0}}} \leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{\color{blue}{9 \cdot a - 3.0}}}\]
    8.5
  8. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot rand}{\sqrt{9 \cdot a - 3.0}} \leadsto (\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot a - 3.0}}\right) * rand + \left(a - \frac{1.0}{3.0}\right))_*\]
    0.1
  9. Applied final simplification

Original test:


(lambda ((a default) (rand default))
  #:name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))