Average Error: 31.2 → 17.5
Time: 1.4s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.2818911259175852 \cdot 10^{+112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -8.924391194312242 \cdot 10^{-179}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.4464803866898927 \cdot 10^{-220}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.9217655476203442 \cdot 10^{-283}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 1.4514146277768898 \cdot 10^{-239}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 4.7489781145347514 \cdot 10^{-151}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 5.347498414219373 \cdot 10^{+101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.2818911259175852 \cdot 10^{+112}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -8.924391194312242 \cdot 10^{-179}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq -1.4464803866898927 \cdot 10^{-220}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.9217655476203442 \cdot 10^{-283}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 1.4514146277768898 \cdot 10^{-239}:\\
\;\;\;\;\log \left(-im\right)\\

\mathbf{elif}\;re \leq 4.7489781145347514 \cdot 10^{-151}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 5.347498414219373 \cdot 10^{+101}:\\
\;\;\;\;\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 -1.2818911259175852e+112)
   (log (- re))
   (if (<= re -8.924391194312242e-179)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re -1.4464803866898927e-220)
       (log (- re))
       (if (<= re -1.9217655476203442e-283)
         (log (sqrt (+ (* re re) (* im im))))
         (if (<= re 1.4514146277768898e-239)
           (log (- im))
           (if (<= re 4.7489781145347514e-151)
             (log im)
             (if (<= re 5.347498414219373e+101)
               (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 <= -1.2818911259175852e+112) {
		tmp = log(-re);
	} else if (re <= -8.924391194312242e-179) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= -1.4464803866898927e-220) {
		tmp = log(-re);
	} else if (re <= -1.9217655476203442e-283) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 1.4514146277768898e-239) {
		tmp = log(-im);
	} else if (re <= 4.7489781145347514e-151) {
		tmp = log(im);
	} else if (re <= 5.347498414219373e+101) {
		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 5 regimes

  2. if re < -1.2818911259175852e112 or -8.9243911943122418e-179 < re < -1.4464803866898927e-220

    1. Initial program 50.6

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

    2. Taylor expanded around -inf 14.8

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

    3. Simplified14.8

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

    if -1.2818911259175852e112 < re < -8.9243911943122418e-179 or -1.4464803866898927e-220 < re < -1.92176554762034421e-283 or 4.7489781145347514e-151 < re < 5.34749841421937321e101

    1. Initial program 17.1

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

    if -1.92176554762034421e-283 < re < 1.4514146277768898e-239

    1. Initial program 29.3

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

    2. Taylor expanded around -inf 30.9

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

    3. Simplified30.9

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

    if 1.4514146277768898e-239 < re < 4.7489781145347514e-151

    1. Initial program 28.4

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

    2. Taylor expanded around 0 36.3

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

    if 5.34749841421937321e101 < re

    1. Initial program 52.2

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

    2. Taylor expanded around inf 7.6

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2818911259175852 \cdot 10^{+112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -8.924391194312242 \cdot 10^{-179}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.4464803866898927 \cdot 10^{-220}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.9217655476203442 \cdot 10^{-283}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 1.4514146277768898 \cdot 10^{-239}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 4.7489781145347514 \cdot 10^{-151}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 5.347498414219373 \cdot 10^{+101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2021009 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))