Average Error: 30.8 → 16.4
Time: 5.5s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.1823101031022743 \cdot 10^{+85}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 8.838551156450988 \cdot 10^{+132}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -2.1823101031022743e+85

    1. Initial program 48.3

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

    2. Taylor expanded around -inf 9.4

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

    3. Applied simplify9.4

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

    if -2.1823101031022743e+85 < re < 8.838551156450988e+132

    1. Initial program 20.5

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

    if 8.838551156450988e+132 < re

    1. Initial program 56.3

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

    2. Taylor expanded around inf 6.3

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

Runtime

Time bar (total: 5.5s)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))