Average Error: 0.1 → 0.2
Time: 41.8s
Precision: 64
Internal Precision: 576
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{1}{\frac{\sqrt{9}}{rand \cdot \sqrt{a - \frac{1.0}{3.0}}}}\]

Error

Bits error versus a

Bits error versus rand

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqrt0.1

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

  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

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

  9. Applied fma-udef0.1

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

  11. Applied associate-*l/0.1

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

  12. Applied associate-*l/0.2

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

  13. Simplified0.2

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

  15. Applied clear-num0.2

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

  16. Final simplification0.2

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

Runtime

Time bar (total: 41.8s)Debug logProfile

herbie shell --seed 2018230 +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))))