Average Error: 31.1 → 16.7
Time: 6.1s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -8.959415571014556 \cdot 10^{+115}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -2.1252318479467404 \cdot 10^{-162}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le -3.868244150438327 \cdot 10^{-189}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.3508184783623033 \cdot 10^{+85}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -8.959415571014556e+115

    1. Initial program 52.8

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

    2. Taylor expanded around inf 7.0

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

    if -8.959415571014556e+115 < (- re) < -2.1252318479467404e-162 or -3.868244150438327e-189 < (- re) < 1.3508184783623033e+85

    1. Initial program 20.8

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

    if -2.1252318479467404e-162 < (- re) < -3.868244150438327e-189

    1. Initial program 26.5

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

    2. Taylor expanded around 0 36.7

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

    if 1.3508184783623033e+85 < (- re)

    1. Initial program 47.1

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

    2. Taylor expanded around -inf 9.3

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

    3. Applied simplify9.3

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

Runtime

Time bar (total: 6.1s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))