Average Error: 30.5 → 17.8
Time: 7.0s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -5.922425173464257 \cdot 10^{+53}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 5020947.316304741:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -5.922425173464257e+53

    1. Initial program 44.2

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

    2. Taylor expanded around inf 11.4

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

    if -5.922425173464257e+53 < (- re) < 5020947.316304741

    1. Initial program 22.1

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

    if 5020947.316304741 < (- re)

    1. Initial program 37.9

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

    2. Taylor expanded around -inf 13.3

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

    3. Applied simplify13.3

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

Runtime

Time bar (total: 7.0s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))