Average Error: 31.2 → 17.5
Time: 7.9s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.5646458293890962 \cdot 10^{+101}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -1.2737350136312853 \cdot 10^{-263}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 8.21968103819049 \cdot 10^{-300}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 0.025734278377922217:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 < -1.5646458293890962e+101

    1. Initial program 50.8

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

    2. Taylor expanded around -inf 8.9

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

    3. Applied simplify8.9

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

    if -1.5646458293890962e+101 < re < -1.2737350136312853e-263 or 8.21968103819049e-300 < re < 0.025734278377922217

    1. Initial program 21.5

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

    if -1.2737350136312853e-263 < re < 8.21968103819049e-300

    1. Initial program 31.7

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

    2. Taylor expanded around 0 30.0

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

    if 0.025734278377922217 < re

    1. Initial program 38.2

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

    2. Taylor expanded around inf 12.8

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

Runtime

Time bar (total: 7.9s)Debug logProfile

herbie shell --seed 2018193 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))