math.abs on complex

Average Error: 31.3 → 17.6

Time: 3.4s

Precision: binary64

Math

\[\]

↓

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Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -3.8911963948436084e148

   1. Initial program 62.5

      \[\]

   2. Taylor expanded around -inf 7.3

      \[\leadsto \]

   3. Simplified 7.3

      \[\leadsto \]

   if -3.8911963948436084e148 < re < -7.0195584571619032e-182 or -1.05288433537404246e-271 < re < 4.84533569762603965e122

   1. Initial program 19.6

      \[\]

   if -7.0195584571619032e-182 < re < -1.05288433537404246e-271

   1. Initial program 32.2

      \[\]

   2. Taylor expanded around 0 34.9

      \[\leadsto \]

   if 4.84533569762603965e122 < re

   1. Initial program 56.3

      \[\]

   2. Taylor expanded around inf 8.9

      \[\leadsto \]
3. Recombined 4 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))