_divideComplex, imaginary part

Average Error: 26.5 → 27.1

Time: 7.4s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.30426463031640256 \cdot 10^{29}:\\ \;\;\;\;\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 2 regimes

2. if y.re < -1.30426463031640256e29

   1. Initial program 34.8

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 34.8

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 34.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   5. Applied times-frac 34.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Taylor expanded around -inf 36.6

      \[\leadsto \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \color{blue}{\left(-1 \cdot x.im\right)}\]

   7. Simplified 36.6

      \[\leadsto \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \color{blue}{\left(-x.im\right)}\]

   if -1.30426463031640256e29 < y.re

   1. Initial program 23.9

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied clear-num 24.1

      \[\leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 27.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.30426463031640256 \cdot 10^{29}:\\ \;\;\;\;\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))