Average Error: 0.1 → 0.1
Time: 2.2m
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[(\left(\frac{rand}{\sqrt{\sqrt{a - \frac{1.0}{3.0}} \cdot \left(9 \cdot \sqrt{a - \frac{1.0}{3.0}}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

Error

Bits error versus a

Bits error versus rand

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied associate-*r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019091 +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))))