Average Error: 0.3 → 0.1
Time: 10.6s
Precision: 64
Internal Precision: 320
\[re \cdot re - im \cdot im\]
↓
\[\left(re - im\right) \cdot \left(im + re\right)\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.3

    \[\left(re \cdot re\right) - \left(im \cdot im\right)\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

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