math.abs on complex

Average Error: 31.2 → 17.3

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.90225451676893219 \cdot 10^{121}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.40575213623674744 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.3980769884343018 \cdot 10^{-199}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6129701241769684 \cdot 10^{115}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -7.90225451676893219e121

   1. Initial program 55.9

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

   2. Taylor expanded around -inf 9.2

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

   3. Simplified 9.2

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

   if -7.90225451676893219e121 < re < -7.40575213623674744e-278 or 4.3980769884343018e-199 < re < 1.6129701241769684e115

   1. Initial program 18.6

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

   if -7.40575213623674744e-278 < re < 4.3980769884343018e-199

   1. Initial program 30.3

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

   2. Taylor expanded around 0 32.9

      \[\leadsto \color{blue}{im}\]

   if 1.6129701241769684e115 < re

   1. Initial program 54.3

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

   2. Taylor expanded around inf 9.3

      \[\leadsto \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.90225451676893219 \cdot 10^{121}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.40575213623674744 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.3980769884343018 \cdot 10^{-199}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6129701241769684 \cdot 10^{115}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))