Error

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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. Initial simplification0.1

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

4. Applied sqrt-prod0.1

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

5. Simplified0.1

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

7. Applied add-sqr-sqrt0.1

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

8. Applied times-frac0.1

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

9. Simplified0.1

   \[\leadsto (\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot \color{blue}{1}\right) \cdot rand + \left(a - \frac{1.0}{3.0}\right))_*\]
10. Using strategy rm

11. Applied fma-udef0.1

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

13. Applied associate-*l/0.1

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

14. Applied associate-*l/0.2

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

15. Simplified0.2

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

16. Final simplification0.2

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

Reproduce

herbie shell --seed 2018360 +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))))

Details

Time bar (total: 15.8s)Debug log