Average Error: 31.6 → 17.8
Time: 1.3s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.0442807027615777 \cdot 10^{+40}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.748570756471217 \cdot 10^{-205}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 1.3080807554136465 \cdot 10^{-270}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 2.616515491898096 \cdot 10^{+82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.0442807027615777 \cdot 10^{+40}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.748570756471217 \cdot 10^{-205}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 1.3080807554136465 \cdot 10^{-270}:\\
\;\;\;\;\log \left(-im\right)\\

\mathbf{elif}\;re \leq 2.616515491898096 \cdot 10^{+82}:\\
\;\;\;\;\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 -1.0442807027615777e+40)
   (log (- re))
   (if (<= re -1.748570756471217e-205)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 1.3080807554136465e-270)
       (log (- im))
       (if (<= re 2.616515491898096e+82)
         (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 <= -1.0442807027615777e+40) {
		tmp = log(-re);
	} else if (re <= -1.748570756471217e-205) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 1.3080807554136465e-270) {
		tmp = log(-im);
	} else if (re <= 2.616515491898096e+82) {
		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 4 regimes

  2. if re < -1.04428070276157775e40

    1. Initial program 43.6

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

    2. Taylor expanded around -inf 11.6

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

    3. Simplified11.6

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

    if -1.04428070276157775e40 < re < -1.7485707564712169e-205 or 1.3080807554136465e-270 < re < 2.61651549189809592e82

    1. Initial program 20.2

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

    if -1.7485707564712169e-205 < re < 1.3080807554136465e-270

    1. Initial program 30.5

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

    2. Taylor expanded around -inf 32.9

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

    3. Simplified32.9

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

    if 2.61651549189809592e82 < re

    1. Initial program 48.3

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

    2. Taylor expanded around inf 9.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.0442807027615777 \cdot 10^{+40}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.748570756471217 \cdot 10^{-205}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 1.3080807554136465 \cdot 10^{-270}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 2.616515491898096 \cdot 10^{+82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))