Average Error: 0.1 → 0.1
Time: 3.1m
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[(\left(\frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{3}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

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 *-commutative0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied associate-/r*0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019088 +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))))