math.abs on complex

Average Error: 31.0 → 17.6

Time: 1.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.3287772795912036 \cdot 10^{57}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.5384845989707597 \cdot 10^{108}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.3287772795912036e57

   1. Initial program 45.5

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

   2. Taylor expanded around -inf 12.3

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

   3. Simplified 12.3

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

   if -2.3287772795912036e57 < re < 7.5384845989707597e108

   1. Initial program 21.0

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

   if 7.5384845989707597e108 < re

   1. Initial program 53.0

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

   2. Taylor expanded around inf 10.6

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.3287772795912036 \cdot 10^{57}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.5384845989707597 \cdot 10^{108}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

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