math.abs on complex

Average Error: 31.7 → 17.3

Time: 1.5s

Precision: binary64

Math

\[\sqrt{re \cdot re + im \cdot im}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.40954284696244225 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -5.7956671044621403e140

   1. Initial program 61.2

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around -inf 7.8

      \[\leadsto \color{blue}{-1 \cdot re}\]

   3. Simplified 7.8

      \[\leadsto \color{blue}{-re}\]

   if -5.7956671044621403e140 < re < 2.40954284696244225e95

   1. Initial program 21.1

      \[\sqrt{re \cdot re + im \cdot im}\]

   if 2.40954284696244225e95 < re

   1. Initial program 50.1

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{re}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.40954284696244225 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))