Average Error: 31.7 → 18.2
Time: 12.5s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.130257577176994 \cdot 10^{+140}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 1.1347647040549314 \cdot 10^{-245}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 6.107346154734439 \cdot 10^{-166}:\\ \;\;\;\;\log re\\ \mathbf{if}\;re \le 1.7956000393992656 \cdot 10^{+141}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -2.130257577176994e+140

    1. Initial program 58.1

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

    2. Taylor expanded around -inf 6.9

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

    3. Applied simplify6.9

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

    if -2.130257577176994e+140 < re < 1.1347647040549314e-245 or 6.107346154734439e-166 < re < 1.7956000393992656e+141

    1. Initial program 20.0

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

    if 1.1347647040549314e-245 < re < 6.107346154734439e-166 or 1.7956000393992656e+141 < re

    1. Initial program 50.8

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

    2. Taylor expanded around inf 20.1

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 12.5s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))