Average Error: 30.0 → 13.9
Time: 18.9s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.5730411458898 \cdot 10^{+118}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le 2.4528332963254056 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 4.8611610197161825 \cdot 10^{-177}:\\ \;\;\;\;re\\ \mathbf{if}\;re \le 3.214839002159321 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -9.5730411458898e+118

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 0

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

    3. Applied simplify0

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

    if -9.5730411458898e+118 < re < 2.4528332963254056e-195 or 4.8611610197161825e-177 < re < 3.214839002159321e+154

    1. Initial program 19.6

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

    if 2.4528332963254056e-195 < re < 4.8611610197161825e-177 or 3.214839002159321e+154 < re

    1. Initial program 59.7

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

    2. Taylor expanded around inf 0

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

Runtime

Time bar (total: 18.9s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))