Average Error: 30.6 → 17.0
Time: 5.9s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.022303069019363 \cdot 10^{+86}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -8.887684374081743 \cdot 10^{-243}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.5641765494886538 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.5079096169475686 \cdot 10^{+106}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -1.022303069019363e+86

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 9.2

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

    3. Applied simplify9.2

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

    if -1.022303069019363e+86 < re < -8.887684374081743e-243 or 1.5641765494886538e-250 < re < 1.5079096169475686e+106

    1. Initial program 18.9

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

    if -8.887684374081743e-243 < re < 1.5641765494886538e-250

    1. Initial program 31.1

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

    2. Taylor expanded around 0 34.4

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

    if 1.5079096169475686e+106 < re

    1. Initial program 51.6

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

    2. Taylor expanded around inf 8.4

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

Runtime

Time bar (total: 5.9s)Debug logProfile

herbie shell --seed 2018207 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))