\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
Test:
Octave 3.8, oct_fill_randg
Bits:
128 bits
Bits error versus a
Bits error versus rand
Time: 20.1 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9}}\right)\)
  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 *-un-lft-identity to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{red}{\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)} \cdot rand\right)\]
    0.1
  4. Applied associate-*l* to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{red}{\left(1 \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{1 \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)}\right)\]
    0.1
  5. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \color{red}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \color{blue}{\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}}\right)\]
    0.1
  6. Using strategy rm
    0.1
  7. Applied sqrt-prod to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \frac{rand}{\color{red}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \frac{rand}{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}}}\right)\]
    0.1
  8. Applied associate-/r* to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \color{red}{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}}}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + 1 \cdot \color{blue}{\frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9}}}\right)\]
    0.1
  9. Removed slow pow expressions

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))))