Average Error: 30.4 → 16.6
Time: 5.6s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.604711439938558 \cdot 10^{+138}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 6.001406428114404 \cdot 10^{+74}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -3.604711439938558e+138

    1. Initial program 58.0

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

    2. Taylor expanded around inf 7.0

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

    if -3.604711439938558e+138 < (- re) < 6.001406428114404e+74

    1. Initial program 20.6

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

    if 6.001406428114404e+74 < (- re)

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 8.6

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

    3. Applied simplify8.6

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

Runtime

Time bar (total: 5.6s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))