Average Error: 30.5 → 16.9
Time: 19.6s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.6674045038045406 \cdot 10^{+136}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -2.9946926564712866 \cdot 10^{-251}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 2.987411589072281 \cdot 10^{-235}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 2.5257366757167287 \cdot 10^{+126}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -3.6674045038045406e+136

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 7.0

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

    3. Applied simplify7.0

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

    if -3.6674045038045406e+136 < re < -2.9946926564712866e-251 or 2.987411589072281e-235 < re < 2.5257366757167287e+126

    1. Initial program 18.4

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

    if -2.9946926564712866e-251 < re < 2.987411589072281e-235

    1. Initial program 30.6

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

    2. Taylor expanded around 0 33.1

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

    if 2.5257366757167287e+126 < re

    1. Initial program 55.7

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

    2. Taylor expanded around inf 8.1

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

Runtime

Time bar (total: 19.6s)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))