Average Error: 0.3 → 0.3
Time: 20.7s
Precision: 64
Internal Precision: 320
\[\left(d1 \cdot d2 + \left(d3 + 5\right) \cdot d1\right) + d1 \cdot 32\]
↓
\[d1 \cdot \left(\left(d2 + 5\right) + d3\right) + d1 \cdot 32\]

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Derivation

  1. Initial program 0.3

    \[\frac{\left(\frac{\left(d1 \cdot d2\right)}{\left(\left(\frac{d3}{\left(real->posit(5)\right)}\right) \cdot d1\right)}\right)}{\left(d1 \cdot \left(real->posit(32)\right)\right)}\]
  2. Using strategy rm

  3. Applied *-commutative0.3

    \[\leadsto \frac{\left(\frac{\left(d1 \cdot d2\right)}{\color{blue}{\left(d1 \cdot \left(\frac{d3}{\left(real->posit(5)\right)}\right)\right)}}\right)}{\left(d1 \cdot \left(real->posit(32)\right)\right)}\]

  4. Applied distribute-lft-out0.3

    \[\leadsto \frac{\color{blue}{\left(d1 \cdot \left(\frac{d2}{\left(\frac{d3}{\left(real->posit(5)\right)}\right)}\right)\right)}}{\left(d1 \cdot \left(real->posit(32)\right)\right)}\]
  5. Using strategy rm

  6. Applied +-commutative0.3

    \[\leadsto \frac{\left(d1 \cdot \left(\frac{d2}{\color{blue}{\left(\frac{\left(real->posit(5)\right)}{d3}\right)}}\right)\right)}{\left(d1 \cdot \left(real->posit(32)\right)\right)}\]

  7. Applied associate-+r+0.3

    \[\leadsto \frac{\left(d1 \cdot \color{blue}{\left(\frac{\left(\frac{d2}{\left(real->posit(5)\right)}\right)}{d3}\right)}\right)}{\left(d1 \cdot \left(real->posit(32)\right)\right)}\]

  8. Final simplification0.3

    \[\leadsto d1 \cdot \left(\left(d2 + 5\right) + d3\right) + d1 \cdot 32\]

Reproduce

herbie shell --seed 2019072 
(FPCore (d1 d2 d3)
  :name "FastMath dist3"
  (+.p16 (+.p16 (*.p16 d1 d2) (*.p16 (+.p16 d3 (real->posit16 5)) d1)) (*.p16 d1 (real->posit16 32))))