Average Error: 31.2 → 18.3
Time: 6.8s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0540904420404546 \cdot 10^{+59}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -2.0260486960111094 \cdot 10^{-255}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 4.973278755616167 \cdot 10^{-139}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.8441144200407667 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -1.0540904420404546e+59

    1. Initial program 43.7

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

    2. Taylor expanded around -inf 10.3

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

    3. Applied simplify10.3

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

    if -1.0540904420404546e+59 < re < -2.0260486960111094e-255 or 4.973278755616167e-139 < re < 1.8441144200407667e+81

    1. Initial program 19.0

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

    if -2.0260486960111094e-255 < re < 4.973278755616167e-139

    1. Initial program 29.0

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

    2. Taylor expanded around 0 33.6

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

    if 1.8441144200407667e+81 < re

    1. Initial program 47.9

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

    2. Taylor expanded around inf 9.9

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

Runtime

Time bar (total: 6.8s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))