Average Error: 31.0 → 16.5
Time: 5.1s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.3556794003074475 \cdot 10^{+95}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 3.21779640511932 \cdot 10^{+101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -1.3556794003074475e+95

    1. Initial program 49.0

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

    2. Taylor expanded around inf 8.5

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

    if -1.3556794003074475e+95 < (- re) < 3.21779640511932e+101

    1. Initial program 20.9

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

    if 3.21779640511932e+101 < (- re)

    1. Initial program 50.7

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

    2. Taylor expanded around -inf 8.1

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

    3. Applied simplify8.1

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

Runtime

Time bar (total: 5.1s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))