math.log/1 on complex, real part

Average Error: 31.9 → 17.1

Time: 1.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.25588737841384258 \cdot 10^{93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -5.7956671044621403e140

   1. Initial program 61.2

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

   2. Taylor expanded around -inf 6.5

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

   3. Simplified 6.5

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

   if -5.7956671044621403e140 < re < 6.25588737841384258e93

   1. Initial program 21.4

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

   if 6.25588737841384258e93 < re

   1. Initial program 49.7

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

   2. Taylor expanded around inf 8.8

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.25588737841384258 \cdot 10^{93}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))