Average Error: 30.8 → 16.7
Time: 18.8s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.5444508077594253 \cdot 10^{+135}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -1.6471939554857127 \cdot 10^{-238}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.3032773090778938 \cdot 10^{-265}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 3.4177957375684004 \cdot 10^{+77}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -2.5444508077594253e+135

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 7.0

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

    3. Applied simplify7.0

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

    if -2.5444508077594253e+135 < re < -1.6471939554857127e-238 or 1.3032773090778938e-265 < re < 3.4177957375684004e+77

    1. Initial program 19.1

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

    if -1.6471939554857127e-238 < re < 1.3032773090778938e-265

    1. Initial program 29.7

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

    2. Taylor expanded around 0 32.9

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

    if 3.4177957375684004e+77 < re

    1. Initial program 45.8

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

    2. Taylor expanded around inf 9.4

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

Runtime

Time bar (total: 18.8s)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))