Average Error: 31.4 → 10.7
Time: 28.1s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.731779904684069 \cdot 10^{+91}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 2.2531375296527453 \cdot 10^{-307}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.099940540275573 \cdot 10^{-139}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 1.0069529393109037 \cdot 10^{+116}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -4.731779904684069e+91

    1. Initial program 50.0

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

    2. Taylor expanded around -inf 0

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

    3. Applied simplify0

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

    if -4.731779904684069e+91 < re < 2.2531375296527453e-307 or 1.099940540275573e-139 < re < 1.0069529393109037e+116

    1. Initial program 18.8

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

    if 2.2531375296527453e-307 < re < 1.099940540275573e-139

    1. Initial program 29.6

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

    2. Taylor expanded around 0 0

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

    if 1.0069529393109037e+116 < re

    1. Initial program 53.7

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

    2. Taylor expanded around inf 0

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

Runtime

Time bar (total: 28.1s)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))