Average Error: 0.2 → 0.2
Time: 27.0s
Precision: 64
Internal Precision: 320
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1.0}{3.0}\right) + \frac{\left(1 \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

    \[\left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-rgt-in0.2

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

  5. Applied associate-*l/0.2

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

  6. Applied associate-*l/0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))