Average Error: 31.2 → 6.6
Time: 2.4s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.212753865436948 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -5.052123158918657 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.212753865436948 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -5.052123158918657 \cdot 10^{-161}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -6.212753865436948e+153)
   (- re)
   (if (<= re -5.052123158918657e-161)
     (sqrt (+ (* re re) (* im im)))
     (+ im (* 0.5 (* re (/ re im)))))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -6.212753865436948e+153) {
		tmp = -re;
	} else if (re <= -5.052123158918657e-161) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = im + (0.5 * (re * (re / 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

Derivation

  1. Split input into 3 regimes

  2. if re < -6.2127538654369482e153

    1. Initial program 63.8

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 4.2

      \[\leadsto \color{blue}{-1 \cdot re}\]

    if -6.2127538654369482e153 < re < -5.05212315891865689e-161

    1. Initial program 10.3

      \[\sqrt{re \cdot re + im \cdot im}\]

    if -5.05212315891865689e-161 < re

    1. Initial program 32.5

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around 0 7.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{im} + im}\]

    3. Simplified7.0

      \[\leadsto \color{blue}{im + 0.5 \cdot \frac{re \cdot re}{im}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary64_7607.0

      \[\leadsto im + 0.5 \cdot \frac{re \cdot re}{\color{blue}{1 \cdot im}}\]

    6. Applied times-frac_binary64_7664.4

      \[\leadsto im + 0.5 \cdot \color{blue}{\left(\frac{re}{1} \cdot \frac{re}{im}\right)}\]

    7. Simplified4.4

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.212753865436948 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -5.052123158918657 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))