Average Error: 31.5 → 17.9
Time: 1.6s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.43594736247992521 \cdot 10^{37}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -6.3388254617832719 \cdot 10^{-190}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 6.5530138229882175 \cdot 10^{-228}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.596063910090653 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -4.43594736247992521e37

    1. Initial program 43.4

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

    2. Taylor expanded around -inf 12.3

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

    3. Simplified12.3

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

    if -4.43594736247992521e37 < re < -6.3388254617832719e-190 or 6.5530138229882175e-228 < re < 2.596063910090653e101

    1. Initial program 18.9

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

    if -6.3388254617832719e-190 < re < 6.5530138229882175e-228

    1. Initial program 29.7

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

    2. Taylor expanded around 0 33.2

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

    if 2.596063910090653e101 < re

    1. Initial program 51.9

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.43594736247992521 \cdot 10^{37}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -6.3388254617832719 \cdot 10^{-190}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 6.5530138229882175 \cdot 10^{-228}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.596063910090653 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))