Average Error: 31.0 → 17.2
Time: 8.0s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.775545272253287 \cdot 10^{+134}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 1.6435142573322878 \cdot 10^{-32}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 3.23777967045036 \cdot 10^{-25}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.6986955270909667 \cdot 10^{+38}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -3.775545272253287e+134

    1. Initial program 56.1

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

    2. Taylor expanded around inf 6.4

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

    if -3.775545272253287e+134 < (- re) < 1.6435142573322878e-32 or 3.23777967045036e-25 < (- re) < 1.6986955270909667e+38

    1. Initial program 21.5

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

    if 1.6435142573322878e-32 < (- re) < 3.23777967045036e-25

    1. Initial program 13.9

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

    2. Taylor expanded around 0 41.4

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

    if 1.6986955270909667e+38 < (- re)

    1. Initial program 42.9

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

    2. Taylor expanded around -inf 11.1

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

    3. Applied simplify11.1

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

Runtime

Time bar (total: 8.0s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))