math.abs on complex

Average Error: 31.6 → 17.9

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.3408035230771657 \cdot 10^{139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.7866568598534521 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.79783213760014545 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.28820419612964551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -9.3408035230771657e139

   1. Initial program 60.5

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

   2. Taylor expanded around -inf 8.4

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

   3. Simplified 8.4

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

   if -9.3408035230771657e139 < re < -4.7866568598534521e-266 or -2.79783213760014545e-306 < re < 1.28820419612964551e92

   1. Initial program 20.8

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

   if -4.7866568598534521e-266 < re < -2.79783213760014545e-306

   1. Initial program 31.2

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

   2. Taylor expanded around 0 31.5

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

   if 1.28820419612964551e92 < re

   1. Initial program 50.9

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

   2. Taylor expanded around inf 11.5

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.3408035230771657 \cdot 10^{139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.7866568598534521 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.79783213760014545 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.28820419612964551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

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