math.log/1 on complex, real part

Average Error: 31.8 → 17.4

Time: 2.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.8946465989197382 \cdot 10^{113}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.68163499683713152 \cdot 10^{-258}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 8.4037824376923499 \cdot 10^{-209}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.7544658168502362 \cdot 10^{115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -3.8946465989197382e113

   1. Initial program 54.7

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

   2. Taylor expanded around -inf 8.5

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

   3. Simplified 8.5

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

   if -3.8946465989197382e113 < re < -4.68163499683713152e-258 or 8.4037824376923499e-209 < re < 8.7544658168502362e115

   1. Initial program 19.2

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

   if -4.68163499683713152e-258 < re < 8.4037824376923499e-209

   1. Initial program 29.9

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

   2. Taylor expanded around 0 31.9

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

   if 8.7544658168502362e115 < re

   1. Initial program 55.0

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

   2. Taylor expanded around inf 8.3

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

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.8946465989197382 \cdot 10^{113}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.68163499683713152 \cdot 10^{-258}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 8.4037824376923499 \cdot 10^{-209}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.7544658168502362 \cdot 10^{115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))