Average Error: 0.1 → 0.1
Time: 5.9s
Precision: 64
Internal Precision: 320
\[re \cdot im + im \cdot re\]
↓
\[im \cdot \left(re + re\right)\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.1

    \[\frac{\left(re \cdot im\right)}{\left(im \cdot re\right)}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\left(im \cdot re\right)}{\left(im \cdot re\right)}}\]
  3. Using strategy rm

  4. Applied p16-distribute-lft-out0.1

    \[\leadsto \color{blue}{im \cdot \left(\frac{re}{re}\right)}\]

  5. Final simplification0.1

    \[\leadsto im \cdot \left(re + re\right)\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  (+.p16 (*.p16 re im) (*.p16 im re)))