Average Error: 32.2 → 18.0
Time: 1.3s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.5452086884385746 \cdot 10^{+75}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.376243184252727 \cdot 10^{-175}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.199094603442337 \cdot 10^{-193}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 7.7571798384429 \cdot 10^{+129}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.5452086884385746 \cdot 10^{+75}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.376243184252727 \cdot 10^{-175}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq -1.199094603442337 \cdot 10^{-193}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 7.7571798384429 \cdot 10^{+129}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -6.5452086884385746e+75)
   (log (- re))
   (if (<= re -1.376243184252727e-175)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re -1.199094603442337e-193)
       (log (- re))
       (if (<= re 7.7571798384429e+129)
         (log (sqrt (+ (* re re) (* im im))))
         (log re))))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -6.5452086884385746e+75) {
		tmp = log(-re);
	} else if (re <= -1.376243184252727e-175) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= -1.199094603442337e-193) {
		tmp = log(-re);
	} else if (re <= 7.7571798384429e+129) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(re);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -6.54520868843857457e75 or -1.376243184252727e-175 < re < -1.1990946034423369e-193

    1. Initial program 46.8

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

    2. Taylor expanded around -inf 12.2

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

    3. Simplified12.2

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

    if -6.54520868843857457e75 < re < -1.376243184252727e-175 or -1.1990946034423369e-193 < re < 7.7571798384428999e129

    1. Initial program 22.1

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

    if 7.7571798384428999e129 < re

    1. Initial program 57.1

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.5452086884385746 \cdot 10^{+75}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.376243184252727 \cdot 10^{-175}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.199094603442337 \cdot 10^{-193}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 7.7571798384429 \cdot 10^{+129}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020299 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))