Average Error: 0.4 → 0.4
Time: 15.5s
Precision: 64
Internal Precision: 320
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  4. Applied *-commutative0.4

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

  6. Applied associate-*r*0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Applied associate--r+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019068 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))