Average Error: 32.2 → 12.3
Time: 21.1s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.165381495034067 \cdot 10^{+96}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 2.4528332963254056 \cdot 10^{-195}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 4.8611610197161825 \cdot 10^{-177}:\\ \;\;\;\;\log re\\ \mathbf{if}\;re \le 1.1415449837537338 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -1.165381495034067e+96

    1. Initial program 49.9

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    2. Taylor expanded around -inf 0

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)}\]

    3. Applied simplify0

      \[\leadsto \color{blue}{\log \left(-re\right)}\]

    if -1.165381495034067e+96 < re < 2.4528332963254056e-195 or 4.8611610197161825e-177 < re < 1.1415449837537338e-08

    1. Initial program 22.0

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    if 2.4528332963254056e-195 < re < 4.8611610197161825e-177 or 1.1415449837537338e-08 < re

    1. Initial program 41.9

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    2. Taylor expanded around inf 0

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

Runtime

Time bar (total: 21.1s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))