math.abs on complex

Average Error: 31.2 → 17.9

Time: 16.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.28654588335027985 \cdot 10^{127}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.34702897925383054 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.32762173516280031 \cdot 10^{-186}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.28654588335027985e127

   1. Initial program 56.2

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

   2. Taylor expanded around -inf 9.0

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

   3. Simplified 9.0

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

   if -1.28654588335027985e127 < re < -1.34702897925383054e-257 or 4.32762173516280031e-186 < re < 3.46617861303372194e36

   1. Initial program 18.5

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

   if -1.34702897925383054e-257 < re < 4.32762173516280031e-186

   1. Initial program 29.9

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

   2. Taylor expanded around 0 33.3

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

   if 3.46617861303372194e36 < re

   1. Initial program 43.9

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

   2. Taylor expanded around inf 12.7

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.28654588335027985 \cdot 10^{127}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.34702897925383054 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.32762173516280031 \cdot 10^{-186}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))