Average Error: 32.6 → 12.6
Time: 26.0s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.060213411501198 \cdot 10^{+97}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 2.447855958190132 \cdot 10^{-195}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.539380988490176 \cdot 10^{-178}:\\ \;\;\;\;\log re\\ \mathbf{if}\;re \le 1.0701985037934348 \cdot 10^{-08}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -3.060213411501198e+97

    1. Initial program 50.4

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

    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 -3.060213411501198e+97 < re < 2.447855958190132e-195 or 1.539380988490176e-178 < re < 1.0701985037934348e-08

    1. Initial program 22.6

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

    if 2.447855958190132e-195 < re < 1.539380988490176e-178 or 1.0701985037934348e-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: 26.0s)Debug log

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