math.abs on complex

Average Error: 31.1 → 17.5

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.36834881583557435 \cdot 10^{131}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.8369579520216515 \cdot 10^{52}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.36834881583557435e131

   1. Initial program 58.1

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

   2. Taylor expanded around -inf 9.6

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

   3. Simplified 9.6

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

   if -1.36834881583557435e131 < re < 4.8369579520216515e52

   1. Initial program 20.8

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

   if 4.8369579520216515e52 < re

   1. Initial program 44.4

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

   2. Taylor expanded around inf 12.5

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.36834881583557435 \cdot 10^{131}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.8369579520216515 \cdot 10^{52}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))