Average Error: 31.2 → 18.6
Time: 7.7s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.1311462515526873 \cdot 10^{+108}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -1.3345219303414604 \cdot 10^{-129}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 7.28790255135744 \cdot 10^{-188}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 6.881791039338204 \cdot 10^{+19}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -1.1311462515526873e+108

    1. Initial program 51.0

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

    2. Taylor expanded around inf 8.6

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

    if -1.1311462515526873e+108 < (- re) < -1.3345219303414604e-129 or 7.28790255135744e-188 < (- re) < 6.881791039338204e+19

    1. Initial program 16.1

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

    if -1.3345219303414604e-129 < (- re) < 7.28790255135744e-188

    1. Initial program 30.8

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

    2. Taylor expanded around 0 34.8

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

    if 6.881791039338204e+19 < (- re)

    1. Initial program 39.7

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

    2. Taylor expanded around -inf 12.6

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

    3. Applied simplify12.6

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

Runtime

Time bar (total: 7.7s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))