Average Error: 0.2 → 0.1
Time: 8.6m
Precision: 64
Internal Precision: 576
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[(\left(\frac{\frac{a - \frac{1.0}{3.0}}{3}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand + \left(a - \frac{1.0}{3.0}\right))_*\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

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

  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 associate-/r*0.1

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

  6. Final simplification0.1

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

Runtime

Time bar (total: 8.6m)Debug logProfile

herbie shell --seed 2018251 +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))))