Average Error: 0.1 → 0.2
Time: 24.8s
Precision: 64
Internal Precision: 320
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{\sqrt{a - \frac{1.0}{3.0}} \cdot rand}{3}\]

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. 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. Applied add-sqr-sqrt0.2

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  10. Applied fma-udef0.1

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

  12. Applied associate-*l/0.1

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

  13. Applied associate-*l/0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Runtime

Time bar (total: 24.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.20.20.00.20%
herbie shell --seed 2018296 +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))))