\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
Test:
Octave 3.8, oct_fill_randg
Bits:
128 bits
Bits error versus a
Bits error versus rand
Time: 25.9 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\((\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{1} \cdot \sqrt{9}}\right) * \left(\frac{rand}{\sqrt{a - 0.3333333333333333}}\right) + \left(a - \frac{1.0}{3.0}\right))_*\)
  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 sqrt-prod to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{red}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}} \cdot rand\right)\]
    0.2
  4. Applied associate-/r* to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{red}{\frac{1}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}} \cdot rand\right)\]
    0.2
  5. Using strategy rm
    0.2
  6. Applied *-un-lft-identity to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{9}}}{\sqrt{\color{red}{a - \frac{1.0}{3.0}}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{9}}}{\sqrt{\color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right)}}} \cdot rand\right)\]
    0.2
  7. Applied sqrt-prod to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{9}}}{\color{red}{\sqrt{1 \cdot \left(a - \frac{1.0}{3.0}\right)}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{9}}}{\color{blue}{\sqrt{1} \cdot \sqrt{a - \frac{1.0}{3.0}}}} \cdot rand\right)\]
    0.2
  8. Applied *-un-lft-identity to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\color{red}{\frac{1}{\sqrt{9}}}}{\sqrt{1} \cdot \sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\color{blue}{1 \cdot \frac{1}{\sqrt{9}}}}{\sqrt{1} \cdot \sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right)\]
    0.2
  9. Applied times-frac to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{red}{\frac{1 \cdot \frac{1}{\sqrt{9}}}{\sqrt{1} \cdot \sqrt{a - \frac{1.0}{3.0}}}} \cdot rand\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right)} \cdot rand\right)\]
    0.2
  10. Applied associate-*l* to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{red}{\left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{1}} \cdot \left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right)}\right)\]
    0.2
  11. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{1}} \cdot \color{red}{\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right)}\right) \leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{1}} \cdot \color{blue}{\frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}}\right)\]
    0.1
  12. Using strategy rm
    0.1
  13. Applied distribute-rgt-in to get
    \[\color{red}{\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{1}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right)} \leadsto \color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)}\]
    0.1
  14. Applied simplify to get
    \[\color{red}{1 \cdot \left(a - \frac{1.0}{3.0}\right)} + \left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) \leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right)} + \left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]
    0.1
  15. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + \color{red}{\left(\frac{1}{\sqrt{1}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)} \leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{1}}}\]
    8.5
  16. Applied taylor to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{1}} \leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - 0.3333333333333333}}}{\sqrt{1}}\]
    8.5
  17. Taylor expanded around 0 to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{\color{red}{a - 0.3333333333333333}}}}{\sqrt{1}} \leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{\color{blue}{a - 0.3333333333333333}}}}{\sqrt{1}}\]
    8.5
  18. Applied simplify to get
    \[\left(a - \frac{1.0}{3.0}\right) + \frac{\frac{\frac{a - \frac{1.0}{3.0}}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - 0.3333333333333333}}}{\sqrt{1}} \leadsto (\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{1} \cdot \sqrt{9}}\right) * \left(\frac{rand}{\sqrt{a - 0.3333333333333333}}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]
    0.1
  19. Applied final simplification
  20. 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))))