\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
Test:
Octave 3.8, oct_fill_randg
Bits:
128 bits
Bits error versus a
Bits error versus rand
Time: 16.1 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\)
  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-rgt-in to get
    \[\color{red}{\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)} \leadsto \color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)}\]
    0.1
  4. Applied simplify to get
    \[\color{red}{1 \cdot \left(a - \frac{1.0}{3.0}\right)} + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right) \leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right)} + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]
    0.1
  5. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + \color{red}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)} \leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right)}\]
    0.1
  6. Using strategy rm
    0.1
  7. Applied div-inv to get
    \[\left(a - \frac{1.0}{3.0}\right) + \color{red}{\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}} \cdot \left(a - \frac{1.0}{3.0}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\left(rand \cdot \frac{1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right)} \cdot \left(a - \frac{1.0}{3.0}\right)\]
    0.1
  8. Applied associate-*l* to get
    \[\left(a - \frac{1.0}{3.0}\right) + \color{red}{\left(rand \cdot \frac{1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)} \leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{rand \cdot \left(\frac{1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}\]
    0.1
  9. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + rand \cdot \color{red}{\left(\frac{1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right)\right)} \leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \color{blue}{\frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}}\]
    0.1

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))))