math.log/2 on complex, real part

Average Error: 31.6 → 18.1

Time: 8.9s

Precision: binary64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3796695805914547 \cdot 10^{+97}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.5366008112942153 \cdot 10^{-167}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base}\\ \mathbf{elif}\;re \leq 1.2029630995824736 \cdot 10^{-228}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq 1.1859849874375555 \cdot 10^{+40}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

↓

(FPCore (re im base)
 :precision binary64
 (if (<= re -1.3796695805914547e+97)
   (/ (log (- re)) (log base))
   (if (<= re -1.5366008112942153e-167)
     (/
      (+
       (* (log base) (log (sqrt (+ (* re re) (* im im)))))
       (* (atan2 im re) 0.0))
      (* (log base) (log base)))
     (if (<= re 1.2029630995824736e-228)
       (/ (log im) (log base))
       (if (<= re 1.1859849874375555e+40)
         (/
          (+
           (* (log base) (log (sqrt (+ (* re re) (* im im)))))
           (* (atan2 im re) 0.0))
          (* (log base) (log base)))
         (/ (log re) (log base)))))))

C

double code(double re, double im, double base) {
	return ((log(sqrt((re * re) + (im * im))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

↓

double code(double re, double im, double base) {
	double tmp;
	if (re <= -1.3796695805914547e+97) {
		tmp = log(-re) / log(base);
	} else if (re <= -1.5366008112942153e-167) {
		tmp = ((log(base) * log(sqrt((re * re) + (im * im)))) + (atan2(im, re) * 0.0)) / (log(base) * log(base));
	} else if (re <= 1.2029630995824736e-228) {
		tmp = log(im) / log(base);
	} else if (re <= 1.1859849874375555e+40) {
		tmp = ((log(base) * log(sqrt((re * re) + (im * im)))) + (atan2(im, re) * 0.0)) / (log(base) * log(base));
	} else {
		tmp = log(re) / log(base);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 4 regimes

2. if re < -1.37966e97

   1. Initial program 50.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 50.3

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]

   3. Taylor expanded around -inf 9.7

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

   4. Simplified 9.7

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

   if -1.37966e97 < re < -1.53661e-167 or 1.20297e-228 < re < 1.18597e40

   1. Initial program 18.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   if -1.53661e-167 < re < 1.20297e-228

   1. Initial program 31.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 31.1

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]

   3. Taylor expanded around 0 34.4

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

   if 1.18597e40 < re

   1. Initial program 43.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 43.3

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]

   3. Taylor expanded around inf 11.1

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3796695805914547 \cdot 10^{+97}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.5366008112942153 \cdot 10^{-167}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base}\\ \mathbf{elif}\;re \leq 1.2029630995824736 \cdot 10^{-228}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq 1.1859849874375555 \cdot 10^{+40}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))