Average Error: 30.9 → 17.2
Time: 7.4s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -4.91903656021437 \cdot 10^{+158}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 1.7279058850430404 \cdot 10^{-266}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 6.629342309651773 \cdot 10^{-233}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 6.410411324008574 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -4.91903656021437e+158

    1. Initial program 62.0

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

    2. Taylor expanded around inf 5.7

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

    if -4.91903656021437e+158 < (- re) < 1.7279058850430404e-266 or 6.629342309651773e-233 < (- re) < 6.410411324008574e+81

    1. Initial program 20.8

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

    if 1.7279058850430404e-266 < (- re) < 6.629342309651773e-233

    1. Initial program 31.2

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

    2. Taylor expanded around 0 30.9

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

    if 6.410411324008574e+81 < (- re)

    1. Initial program 46.9

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

    2. Taylor expanded around -inf 9.7

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

    3. Applied simplify9.7

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 7.4s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))