Average Error: 31.5 → 17.5
Time: 2.8s
Precision: binary64
Cost: 14532
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -5.810506997467958 \cdot 10^{+110}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;im \leq -4.996387093174353 \cdot 10^{-297}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;im \leq 3.135452371686857 \cdot 10^{-180}:\\ \;\;\;\;\log \left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)\\ \mathbf{elif}\;im \leq 3.757890946018409 \cdot 10^{+89}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \leq -5.810506997467958 \cdot 10^{+110}:\\
\;\;\;\;\log \left(-im\right)\\

\mathbf{elif}\;im \leq -4.996387093174353 \cdot 10^{-297}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;im \leq 3.135452371686857 \cdot 10^{-180}:\\
\;\;\;\;\log \left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)\\

\mathbf{elif}\;im \leq 3.757890946018409 \cdot 10^{+89}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im -5.810506997467958e+110)
   (log (- im))
   (if (<= im -4.996387093174353e-297)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= im 3.135452371686857e-180)
       (log (- (* (/ (* im im) re) -0.5) re))
       (if (<= im 3.757890946018409e+89)
         (log (sqrt (+ (* re re) (* im im))))
         (log im))))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= -5.810506997467958e+110) {
		tmp = log(-im);
	} else if (im <= -4.996387093174353e-297) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (im <= 3.135452371686857e-180) {
		tmp = log((((im * im) / re) * -0.5) - re);
	} else if (im <= 3.757890946018409e+89) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(im);
	}
	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

Alternatives

Alternative 1
Error48.6
Cost13952
\[\log \left(\left(re + \frac{im \cdot im}{re} \cdot 0.5\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)\]
Alternative 2
Error48.3
Cost13952
\[\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)\]
Alternative 3
Error31.5
Cost13248
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Alternative 4
Error47.7
Cost6976
\[\log \left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)\]
Alternative 5
Error47.1
Cost6528
\[\log \left(-im\right)\]
Alternative 6
Error46.2
Cost6528
\[\log \left(-re\right)\]
Alternative 7
Error46.4
Cost6464
\[\log re\]
Alternative 8
Error46.1
Cost6464
\[\log im\]
Alternative 9
Error57.1
Cost64
\[1\]
Alternative 10
Error62.0
Cost64
\[0\]
Alternative 11
Error60.9
Cost64
\[-1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if im < -5.810506997467958e110

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    4. Simplified8.5

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

    if -5.810506997467958e110 < im < -4.9963870931743528e-297 or 3.1354523716868571e-180 < im < 3.7578909460184092e89

    1. Initial program 19.4

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

    2. Simplified19.4

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

    if -4.9963870931743528e-297 < im < 3.1354523716868571e-180

    1. Initial program 32.4

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

    2. Taylor expanded around -inf 33.4

      \[\leadsto \log \color{blue}{\left(-\left(re + 0.5 \cdot \frac{{im}^{2}}{re}\right)\right)}\]

    3. Simplified33.4

      \[\leadsto \log \color{blue}{\left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)}\]

    4. Simplified33.4

      \[\leadsto \color{blue}{\log \left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)}\]

    if 3.7578909460184092e89 < im

    1. Initial program 50.1

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

    2. Taylor expanded around 0 8.8

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

    3. Simplified8.8

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.810506997467958 \cdot 10^{+110}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;im \leq -4.996387093174353 \cdot 10^{-297}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;im \leq 3.135452371686857 \cdot 10^{-180}:\\ \;\;\;\;\log \left(\frac{im \cdot im}{re} \cdot -0.5 - re\right)\\ \mathbf{elif}\;im \leq 3.757890946018409 \cdot 10^{+89}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))