Average Error: 0.1 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\left(1 - \frac{1}{2}\right)}}{\frac{\sqrt{9}}{rand}}\]

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. Using strategy rm

  3. Applied pow1/20.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{1}{2}}}} \cdot rand\right)\]

  4. Applied pow-flip0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\left(-\frac{1}{2}\right)}} \cdot rand\right)\]
  5. Using strategy rm

  6. Applied unpow-prod-down0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\left({9}^{\left(-\frac{1}{2}\right)} \cdot {\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)}\right)} \cdot rand\right)\]

  7. Applied associate-*l*0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{{9}^{\left(-\frac{1}{2}\right)} \cdot \left({\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)} \cdot rand\right)}\right)\]
  8. Using strategy rm

  9. Applied neg-sub00.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {9}^{\color{blue}{\left(0 - \frac{1}{2}\right)}} \cdot \left({\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)} \cdot rand\right)\right)\]

  10. Applied pow-sub0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\frac{{9}^{0}}{{9}^{\frac{1}{2}}}} \cdot \left({\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)} \cdot rand\right)\right)\]

  11. Applied associate-*l/0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\frac{{9}^{0} \cdot \left({\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)} \cdot rand\right)}{{9}^{\frac{1}{2}}}}\right)\]

  12. Simplified0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\color{blue}{rand \cdot {\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)}}}{{9}^{\frac{1}{2}}}\right)\]
  13. Using strategy rm

  14. Applied distribute-rgt-in0.1

    \[\leadsto \color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right) + \frac{rand \cdot {\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)}}{{9}^{\frac{1}{2}}} \cdot \left(a - \frac{1.0}{3.0}\right)}\]

  15. Simplified0.1

    \[\leadsto 1 \cdot \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{{\left(a - \frac{1.0}{3.0}\right)}^{\left(1 - \frac{1}{2}\right)}}{\frac{\sqrt{9}}{rand}}}\]

  16. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\left(1 - \frac{1}{2}\right)}}{\frac{\sqrt{9}}{rand}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018230 
(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))))