Average Error: 31.3 → 17.9
Time: 5.9s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -7.440968527251743 \cdot 10^{+68}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -4.7713816172072814 \cdot 10^{-213}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 1.9902505349126515 \cdot 10^{-239}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.8961716390801075 \cdot 10^{+88}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -7.440968527251743e+68

    1. Initial program 45.0

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

    2. Taylor expanded around inf 10.3

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

    if -7.440968527251743e+68 < (- re) < -4.7713816172072814e-213 or 1.9902505349126515e-239 < (- re) < 1.8961716390801075e+88

    1. Initial program 19.8

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

    if -4.7713816172072814e-213 < (- re) < 1.9902505349126515e-239

    1. Initial program 30.2

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

    2. Taylor expanded around 0 34.3

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

    if 1.8961716390801075e+88 < (- re)

    1. Initial program 49.3

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

    2. Taylor expanded around -inf 8.5

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

    3. Applied simplify8.5

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 5.9s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))