Average Error: 30.2 → 16.4
Time: 3.4s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.350481916791837 \cdot 10^{+87}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 1.9977783303608204 \cdot 10^{+115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -5.350481916791837e+87

    1. Initial program 48.0

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

    2. Taylor expanded around -inf 8.5

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

    3. Applied simplify8.5

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

    if -5.350481916791837e+87 < re < 1.9977783303608204e+115

    1. Initial program 20.6

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

    if 1.9977783303608204e+115 < re

    1. Initial program 52.2

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

    2. Taylor expanded around inf 7.0

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

Runtime

Time bar (total: 3.4s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))