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

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

4. Applied fma-udef0.1

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

6. Applied sqrt-prod0.1

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

7. Applied associate-/r*0.1

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

9. Applied add-sqr-sqrt0.1

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

10. Applied associate-*r*0.1

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

11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(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))))