Average Error: 31.2 → 17.9
Time: 1.8s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -5.2032752033560465 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.0928214580192916 \cdot 10^{-154}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.4695374740654386 \cdot 10^{-180}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 8.833768332896172 \cdot 10^{+88}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -5.2032752033560465 \cdot 10^{+103}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.0928214580192916 \cdot 10^{-154}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 2.4695374740654386 \cdot 10^{-180}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 8.833768332896172 \cdot 10^{+88}:\\
\;\;\;\;\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 -5.2032752033560465e+103)
   (log (- re))
   (if (<= re -1.0928214580192916e-154)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 2.4695374740654386e-180)
       (log im)
       (if (<= re 8.833768332896172e+88)
         (log (sqrt (+ (* re re) (* im im))))
         (log re))))))
double code(double re, double im) {
	return ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
}
↓
double code(double re, double im) {
	double VAR;
	if ((re <= -5.2032752033560465e+103)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= -1.0928214580192916e-154)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 2.4695374740654386e-180)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 8.833768332896172e+88)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					VAR_3 = ((double) log(re));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -5.2032752033560465e103

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -5.2032752033560465e103 < re < -1.09282145801929164e-154 or 2.46953747406543856e-180 < re < 8.8337683328961723e88

    1. Initial program 16.2

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

    if -1.09282145801929164e-154 < re < 2.46953747406543856e-180

    1. Initial program 31.0

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

    2. Taylor expanded around 0 34.0

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

    if 8.8337683328961723e88 < re

    1. Initial program 49.4

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

    2. Taylor expanded around inf 9.3

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2032752033560465 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.0928214580192916 \cdot 10^{-154}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.4695374740654386 \cdot 10^{-180}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 8.833768332896172 \cdot 10^{+88}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))