math.abs on complex

Average Error: 31.7 → 17.8

Time: 2.5s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.4364574278384972 \cdot 10^{-206}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.69108546213632265e113

   1. Initial program 54.7

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

   2. Taylor expanded around -inf 10.1

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

   3. Simplified 10.1

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

   if -4.69108546213632265e113 < re < -7.88936342124094937e-297 or 3.4364574278384972e-206 < re < 4.4095657239434897e116

   1. Initial program 19.6

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

   if -7.88936342124094937e-297 < re < 3.4364574278384972e-206

   1. Initial program 29.4

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

   2. Taylor expanded around 0 32.0

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

   if 4.4095657239434897e116 < re

   1. Initial program 55.0

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.4364574278384972 \cdot 10^{-206}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))