Average Error: 30.7 → 16.8
Time: 7.3s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.0449150866979644 \cdot 10^{+64}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -1.8131243705185465 \cdot 10^{-306}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.0617321238763363 \cdot 10^{-273}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.281885041742443 \cdot 10^{+90}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -3.0449150866979644e+64

    1. Initial program 45.0

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

    2. Taylor expanded around -inf 9.9

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

    3. Applied simplify9.9

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

    if -3.0449150866979644e+64 < re < -1.8131243705185465e-306 or 1.0617321238763363e-273 < re < 1.281885041742443e+90

    1. Initial program 20.5

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

    if -1.8131243705185465e-306 < re < 1.0617321238763363e-273

    1. Initial program 32.8

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

    2. Taylor expanded around 0 34.4

      \[\leadsto \log \color{blue}{im}\]

    if 1.281885041742443e+90 < re

    1. Initial program 48.3

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

    2. Taylor expanded around inf 8.8

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

Runtime

Time bar (total: 7.3s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))