math.log/1 on complex, real part

Average Error: 32.4 → 18.2

Time: 1.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.30931441283709317 \cdot 10^{111}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.87107645438188905 \cdot 10^{-266}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.3130803964525389 \cdot 10^{-192}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.25643977270018092 \cdot 10^{27}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -9.30931441283709317e111

   1. Initial program 54.2

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

   2. Taylor expanded around -inf 8.1

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

   3. Simplified 8.1

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

   if -9.30931441283709317e111 < re < 3.87107645438188905e-266 or 4.3130803964525389e-192 < re < 4.25643977270018092e27

   1. Initial program 21.7

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

   if 3.87107645438188905e-266 < re < 4.3130803964525389e-192

   1. Initial program 32.1

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

   2. Taylor expanded around 0 32.8

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

   if 4.25643977270018092e27 < re

   1. Initial program 42.4

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

   2. Taylor expanded around inf 13.2

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.30931441283709317 \cdot 10^{111}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.87107645438188905 \cdot 10^{-266}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.3130803964525389 \cdot 10^{-192}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.25643977270018092 \cdot 10^{27}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))