Average Error: 30.6 → 18.0
Time: 7.0s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -4.355043212826246 \cdot 10^{+155}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -7.719760126663409 \cdot 10^{-204}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 1.6282427533110736 \cdot 10^{-233}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 4.1983888306306506 \cdot 10^{+76}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -4.355043212826246e+155

    1. Initial program 62.0

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

    2. Taylor expanded around inf 6.9

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

    if -4.355043212826246e+155 < (- re) < -7.719760126663409e-204 or 1.6282427533110736e-233 < (- re) < 4.1983888306306506e+76

    1. Initial program 18.4

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

    if -7.719760126663409e-204 < (- re) < 1.6282427533110736e-233

    1. Initial program 29.9

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

    2. Taylor expanded around 0 35.3

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

    if 4.1983888306306506e+76 < (- re)

    1. Initial program 46.4

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

    2. Taylor expanded around -inf 10.3

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

    3. Applied simplify10.3

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 7.0s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))