Average Error: 30.5 → 18.3
Time: 10.3s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.441828457252553 \cdot 10^{+147}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le 1.1347647040549314 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 6.107346154734439 \cdot 10^{-166}:\\ \;\;\;\;re\\ \mathbf{if}\;re \le 1.686931697481935 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -7.441828457252553e+147

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 7.5

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

    3. Applied simplify7.5

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

    if -7.441828457252553e+147 < re < 1.1347647040549314e-245 or 6.107346154734439e-166 < re < 1.686931697481935e+141

    1. Initial program 19.3

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

    if 1.1347647040549314e-245 < re < 6.107346154734439e-166 or 1.686931697481935e+141 < re

    1. Initial program 48.7

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

    2. Taylor expanded around inf 21.7

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

Runtime

Time bar (total: 10.3s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))