Average Error: 32.2 → 7.0
Time: 2.1s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \frac{re}{\frac{im}{re}}\right)\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(im + 0.5 \cdot \frac{re}{\frac{im}{re}}\right)\\

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -7.143276769119883e+148)
   (log (- re))
   (if (<= re -7.144782444788284e-138)
     (log (sqrt (+ (* re re) (* im im))))
     (log (+ im (* 0.5 (/ re (/ im re))))))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -7.143276769119883e+148) {
		tmp = log(-re);
	} else if (re <= -7.144782444788284e-138) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(im + (0.5 * (re / (im / 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 < -7.1432767691198832e148

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 3.4

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

    3. Simplified3.4

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

    if -7.1432767691198832e148 < re < -7.1447824447882838e-138

    1. Initial program 11.1

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

    if -7.1447824447882838e-138 < re

    1. Initial program 32.6

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

    2. Taylor expanded around 0 7.9

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

    3. Simplified7.9

      \[\leadsto \log \color{blue}{\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)}\]
    4. Using strategy rm

    5. Applied associate-/l*_binary645.5

      \[\leadsto \log \left(im + 0.5 \cdot \color{blue}{\frac{re}{\frac{im}{re}}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.143276769119883 \cdot 10^{+148}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -7.144782444788284 \cdot 10^{-138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \frac{re}{\frac{im}{re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))