Average Error: 31.0 → 16.7
Time: 8.8s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.310955611845655 \cdot 10^{+109}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 7.832763149879082 \cdot 10^{-290}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 7.184568937459825 \cdot 10^{-264}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 4.808391128987194 \cdot 10^{+80}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -5.310955611845655e+109

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.1

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

    3. Applied simplify8.1

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

    if -5.310955611845655e+109 < re < 7.832763149879082e-290 or 7.184568937459825e-264 < re < 4.808391128987194e+80

    1. Initial program 20.6

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

    if 7.832763149879082e-290 < re < 7.184568937459825e-264

    1. Initial program 28.5

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

    2. Taylor expanded around 0 31.3

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

    if 4.808391128987194e+80 < re

    1. Initial program 48.0

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

    2. Taylor expanded around inf 9.2

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

Runtime

Time bar (total: 8.8s)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))