Average Error: 0.1 → 0.1
Time: 56.8s
Precision: 64
Internal Precision: 320
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \sqrt{a - \frac{1.0}{3.0}} \cdot \left(\sqrt{a - \frac{1.0}{3.0}} \cdot \frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1.0}{3.0}}}\right)\]

Error

Bits error versus a

Bits error versus rand

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Results

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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. Using strategy rm

  3. Applied associate-*l/0.1

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

  4. Simplified0.1

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

  6. Applied distribute-rgt-in0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-/r*0.1

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

  11. Applied add-sqr-sqrt0.1

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

  12. Applied associate-*r*0.1

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

  13. Final simplification0.1

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

Runtime

Time bar (total: 56.8s)Debug logProfile

herbie shell --seed 2018225 
(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))))