Average Error: 30.7 → 17.1
Time: 5.5s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.6864484995119144 \cdot 10^{+150}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 3.638105894156943 \cdot 10^{+65}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -1.6864484995119144e+150

    1. Initial program 61.1

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

    2. Taylor expanded around -inf 5.8

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

    3. Applied simplify5.8

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

    if -1.6864484995119144e+150 < re < 3.638105894156943e+65

    1. Initial program 21.0

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

    if 3.638105894156943e+65 < re

    1. Initial program 45.7

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

    2. Taylor expanded around inf 10.9

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

Runtime

Time bar (total: 5.5s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))