Average Error: 30.6 → 17.4
Time: 6.6s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.571051457351973 \cdot 10^{+21}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 2.8128963564335743 \cdot 10^{+106}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -6.571051457351973e+21

    1. Initial program 40.0

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

    2. Taylor expanded around -inf 12.6

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

    3. Applied simplify12.6

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

    if -6.571051457351973e+21 < re < 2.8128963564335743e+106

    1. Initial program 21.5

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

    if 2.8128963564335743e+106 < re

    1. Initial program 52.0

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

    2. Taylor expanded around inf 8.6

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

Runtime

Time bar (total: 6.6s)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))