math.log/1 on complex, real part

Average Error: 31.4 → 17.4

Time: 17.0s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.8794074829285631 \cdot 10^{125}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.82707962304541887 \cdot 10^{-251}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.10205923095853832 \cdot 10^{-190}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -5.8794074829285631e125

   1. Initial program 56.1

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

   2. Taylor expanded around -inf 7.7

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

   3. Simplified 7.7

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

   if -5.8794074829285631e125 < re < -1.82707962304541887e-251 or 4.10205923095853832e-190 < re < 3.46617861303372194e36

   1. Initial program 18.7

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

   if -1.82707962304541887e-251 < re < 4.10205923095853832e-190

   1. Initial program 30.3

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

   2. Taylor expanded around 0 33.2

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

   if 3.46617861303372194e36 < re

   1. Initial program 43.9

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

   2. Taylor expanded around inf 10.8

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

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.8794074829285631 \cdot 10^{125}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.82707962304541887 \cdot 10^{-251}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.10205923095853832 \cdot 10^{-190}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))