Average Error: 31.2 → 17.6
Time: 7.2s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1998496369053764 \cdot 10^{+154}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -5.390990571923889 \cdot 10^{-284}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 4.001925873943323 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.172611339478759 \cdot 10^{+62}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 < -1.1998496369053764e+154

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 7.5

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

    3. Applied simplify7.5

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

    if -1.1998496369053764e+154 < re < -5.390990571923889e-284 or 4.001925873943323e-250 < re < 1.172611339478759e+62

    1. Initial program 20.2

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

    if -5.390990571923889e-284 < re < 4.001925873943323e-250

    1. Initial program 33.3

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

    2. Taylor expanded around 0 33.6

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

    if 1.172611339478759e+62 < re

    1. Initial program 43.7

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

    2. Taylor expanded around inf 10.2

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

Runtime

Time bar (total: 7.2s)Debug logProfile

herbie shell --seed 2018199 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))