Average Error: 30.9 → 18.1
Time: 4.4s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2201915057062602 \cdot 10^{+27}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -6.579510600567167 \cdot 10^{-141}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le -3.615893621001899 \cdot 10^{-249}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.695582353543743 \cdot 10^{+118}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -1.2201915057062602e+27

    1. Initial program 41.6

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

    2. Taylor expanded around -inf 12.1

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

    3. Applied simplify12.1

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

    if -1.2201915057062602e+27 < re < -6.579510600567167e-141 or -3.615893621001899e-249 < re < 1.695582353543743e+118

    1. Initial program 20.1

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

    if -6.579510600567167e-141 < re < -3.615893621001899e-249

    1. Initial program 29.1

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

    2. Taylor expanded around 0 37.7

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

    if 1.695582353543743e+118 < re

    1. Initial program 52.6

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

    2. Taylor expanded around inf 8.3

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

Runtime

Time bar (total: 4.4s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))