Error

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

2. Simplified0.1

   \[\leadsto \color{blue}{rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} + \left(a - \frac{1.0}{3.0}\right)}\]
3. Using strategy rm

4. Applied *-commutative0.1

   \[\leadsto rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{\color{blue}{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}} + \left(a - \frac{1.0}{3.0}\right)\]
5. Using strategy rm

6. Applied sqrt-prod0.1

   \[\leadsto rand \cdot \frac{a - \frac{1.0}{3.0}}{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}}} + \left(a - \frac{1.0}{3.0}\right)\]

7. Applied *-un-lft-identity0.1

   \[\leadsto rand \cdot \frac{\color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right)}}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}} + \left(a - \frac{1.0}{3.0}\right)\]

8. Applied times-frac0.1

   \[\leadsto rand \cdot \color{blue}{\left(\frac{1}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9}}\right)} + \left(a - \frac{1.0}{3.0}\right)\]

9. Applied associate-*r*0.1

   \[\leadsto \color{blue}{\left(rand \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9}}} + \left(a - \frac{1.0}{3.0}\right)\]

10. Simplified0.1

    \[\leadsto \left(rand \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \color{blue}{\frac{a - \frac{1.0}{3.0}}{3}} + \left(a - \frac{1.0}{3.0}\right)\]

11. Final simplification0.1

    \[\leadsto \frac{a - \frac{1.0}{3.0}}{3} \cdot \left(rand \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019093 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))