Average Error: 0.5 → 0.5
Time: 44.3s
Precision: 64
Internal Precision: 320
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[rand \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\right) + 1 \cdot \left(a - \frac{1.0}{3.0}\right)\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.5

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

  3. Applied +-commutative0.5

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

  4. Applied distribute-rgt-in0.5

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

  6. Applied *-commutative0.5

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

  7. Applied associate-*l*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019072 
(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))))