Average Error: 32.1 → 17.6
Time: 1.1s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.8311489939222134 \cdot 10^{+74}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 3.234431479514519 \cdot 10^{+70}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.8311489939222134 \cdot 10^{+74}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 3.234431479514519 \cdot 10^{+70}:\\
\;\;\;\;\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 -2.8311489939222134e+74)
   (log (- re))
   (if (<= re 3.234431479514519e+70)
     (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 <= -2.8311489939222134e+74) {
		tmp = log(-re);
	} else if (re <= 3.234431479514519e+70) {
		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 < -2.83114899392221336e74

    1. Initial program 48.0

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

    2. Taylor expanded around -inf 10.4

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

    3. Simplified10.4

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

    if -2.83114899392221336e74 < re < 3.234431479514519e70

    1. Initial program 22.1

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

    if 3.234431479514519e70 < re

    1. Initial program 48.1

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

    2. Taylor expanded around inf 10.1

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8311489939222134 \cdot 10^{+74}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 3.234431479514519 \cdot 10^{+70}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))