Average Error: 32.2 → 19.1
Time: 1.9s
Precision: binary64
Cost: 15174
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.0797067422693544 \cdot 10^{+85}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -3.466869228507893 \cdot 10^{-172}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \leq -3.283714079027123 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)\\ \mathbf{elif}\;re \leq 2.0246090950680006 \cdot 10^{-222}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 6.851137618913075 \cdot 10^{-132}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 0.0003766592687471088:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.0797067422693544 \cdot 10^{+85}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -3.466869228507893 \cdot 10^{-172}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \leq -3.283714079027123 \cdot 10^{-236}:\\
\;\;\;\;\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)\\

\mathbf{elif}\;re \leq 2.0246090950680006 \cdot 10^{-222}:\\
\;\;\;\;\log \left(-im\right)\\

\mathbf{elif}\;re \leq 6.851137618913075 \cdot 10^{-132}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 0.0003766592687471088:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\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 -6.0797067422693544e+85)
   (log (- re))
   (if (<= re -3.466869228507893e-172)
     (log (sqrt (+ (* im im) (* re re))))
     (if (<= re -3.283714079027123e-236)
       (log
        (-
         (+ im (* 0.5 (/ (* re re) im)))
         (* 0.125 (* re (pow (/ re im) 3.0)))))
       (if (<= re 2.0246090950680006e-222)
         (log (- im))
         (if (<= re 6.851137618913075e-132)
           (log im)
           (if (<= re 0.0003766592687471088)
             (log (sqrt (+ (* im im) (* re re))))
             (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 <= -6.0797067422693544e+85) {
		tmp = log(-re);
	} else if (re <= -3.466869228507893e-172) {
		tmp = log(sqrt((im * im) + (re * re)));
	} else if (re <= -3.283714079027123e-236) {
		tmp = log((im + (0.5 * ((re * re) / im))) - (0.125 * (re * pow((re / im), 3.0))));
	} else if (re <= 2.0246090950680006e-222) {
		tmp = log(-im);
	} else if (re <= 6.851137618913075e-132) {
		tmp = log(im);
	} else if (re <= 0.0003766592687471088) {
		tmp = log(sqrt((im * im) + (re * re)));
	} 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

Alternatives

Alternative 1
Error49.0
Cost13952
\[\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)\]
Alternative 2
Error48.1
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
Error32.2
Cost13248
\[\log \left(\sqrt{im \cdot im + re \cdot re}\right)\]
Alternative 4
Error46.8
Cost6528
\[\log \left(-im\right)\]
Alternative 5
Error46.2
Cost6528
\[\log \left(-re\right)\]
Alternative 6
Error46.9
Cost6464
\[\log re\]
Alternative 7
Error46.0
Cost6464
\[\log im\]
Alternative 8
Error57.0
Cost64
\[1\]
Alternative 9
Error62.0
Cost64
\[0\]
Alternative 10
Error60.9
Cost64
\[-1\]

Error

Derivation

  1. Split input into 6 regimes

  2. if re < -6.0797067422693544e85

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 10.0

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

    3. Simplified10.0

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

    4. Simplified10.0

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

    if -6.0797067422693544e85 < re < -3.4668692285078929e-172 or 6.851137618913075e-132 < re < 3.76659268747108822e-4

    1. Initial program 16.6

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

    2. Simplified16.6

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

    if -3.4668692285078929e-172 < re < -3.2837140790271231e-236

    1. Initial program 31.2

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

    2. Taylor expanded around 0 40.7

      \[\leadsto \log \color{blue}{\left(\left(0.5 \cdot \frac{{re}^{2}}{im} + im\right) - 0.125 \cdot \frac{{re}^{4}}{{im}^{3}}\right)}\]

    3. Simplified35.9

      \[\leadsto \log \color{blue}{\left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left({\left(\frac{re}{im}\right)}^{3} \cdot re\right)\right)}\]

    4. Simplified35.9

      \[\leadsto \color{blue}{\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)}\]

    if -3.2837140790271231e-236 < re < 2.0246090950680006e-222

    1. Initial program 31.3

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

    2. Taylor expanded around -inf 33.3

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

    3. Simplified33.3

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

    4. Simplified33.3

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

    if 2.0246090950680006e-222 < re < 6.851137618913075e-132

    1. Initial program 27.2

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

    2. Taylor expanded around 0 35.0

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

    3. Simplified35.0

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

    if 3.76659268747108822e-4 < re

    1. Initial program 40.8

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

    2. Taylor expanded around inf 13.1

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

    3. Simplified13.1

      \[\leadsto \color{blue}{\log re}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification19.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.0797067422693544 \cdot 10^{+85}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -3.466869228507893 \cdot 10^{-172}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \leq -3.283714079027123 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)\\ \mathbf{elif}\;re \leq 2.0246090950680006 \cdot 10^{-222}:\\ \;\;\;\;\log \left(-im\right)\\ \mathbf{elif}\;re \leq 6.851137618913075 \cdot 10^{-132}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 0.0003766592687471088:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))