Average Error: 29.6 → 16.7
Time: 26.6s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.316015668573352 \cdot 10^{+138}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le -5.706169994009356 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 9.750924739353902 \cdot 10^{-266}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 1.6437527808210546 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -2.316015668573352e+138

    1. Initial program 54.6

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

    2. Taylor expanded around -inf 8.1

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

    3. Applied simplify8.1

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

    if -2.316015668573352e+138 < re < -5.706169994009356e-239 or 9.750924739353902e-266 < re < 1.6437527808210546e+140

    1. Initial program 18.1

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

    if -5.706169994009356e-239 < re < 9.750924739353902e-266

    1. Initial program 28.7

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

    2. Taylor expanded around 0 33.3

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

    if 1.6437527808210546e+140 < re

    1. Initial program 55.1

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

    2. Taylor expanded around inf 8.5

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

Runtime

Time bar (total: 26.6s)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))