Average Error: 30.2 → 17.3
Time: 7.6s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0263210743075056 \cdot 10^{+53}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -7.171146468844302 \cdot 10^{-138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 2.4777204316599815 \cdot 10^{-214}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 7.717561609312647 \cdot 10^{+111}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 < -1.0263210743075056e+53

    1. Initial program 43.2

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

    2. Taylor expanded around -inf 9.8

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

    3. Applied simplify9.8

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

    if -1.0263210743075056e+53 < re < -7.171146468844302e-138 or 2.4777204316599815e-214 < re < 7.717561609312647e+111

    1. Initial program 16.3

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

    if -7.171146468844302e-138 < re < 2.4777204316599815e-214

    1. Initial program 28.5

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

    2. Taylor expanded around 0 33.7

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

    if 7.717561609312647e+111 < re

    1. Initial program 52.1

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

    2. Taylor expanded around inf 8.2

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

Runtime

Time bar (total: 7.6s)Debug logProfile

herbie shell --seed 2018198 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))