math.cube on complex, real part

Average Error: 7.8 → 0.3

Time: 3.4s

Precision: binary64

Math

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

↓

\[\left(x.re + x.im\right) \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(x.re \cdot \left(-\left(\left(x.re + x.im\right) + x.im \cdot 2\right)\right)\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.8
Target	0.3
Herbie	0.3

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

1. Initial program 7.8

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
2. Using strategy rm

3. Applied difference-of-squares 7.8

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

4. Applied associate-*l* 0.3

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

5. Simplified 0.3

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

7. Applied sub-neg 0.3

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

8. Applied distribute-lft-in 0.3

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

9. Applied distribute-lft-in 0.3

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

10. Applied associate--l+ 0.3

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

11. Simplified 0.3

    \[\leadsto \left(x.re + x.im\right) \cdot \left(x.re \cdot x.re\right) + \color{blue}{\left(-x.im\right) \cdot \left(x.re \cdot \left(\left(x.re + x.im\right) + x.im \cdot 2\right)\right)}\]

12. Final simplification 0.3

    \[\leadsto \left(x.re + x.im\right) \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(x.re \cdot \left(-\left(\left(x.re + x.im\right) + x.im \cdot 2\right)\right)\right)\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))