Average Error: 31.9 → 7.2
Time: 1.6s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im} \]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 4.776300407843372 \cdot 10^{-164}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 2.5184939753787586 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 4.776300407843372 \cdot 10^{-164}:\\
\;\;\;\;-re\\

\mathbf{elif}\;im \leq 2.5184939753787586 \cdot 10^{+113}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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


\end{array}

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore modulus (re im)
 :precision binary64
 (if (<= im 4.776300407843372e-164)
   (- re)
   (if (<= im 2.5184939753787586e+113) (sqrt (+ (* re re) (* im im))) im)))
double modulus(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double modulus(double re, double im) {
	double tmp;
	if (im <= 4.776300407843372e-164) {
		tmp = -re;
	} else if (im <= 2.5184939753787586e+113) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = 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 im < 4.7763004078433717e-164

    1. Initial program 32.5

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

    2. Taylor expanded around -inf 5.4

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

    3. Simplified5.4

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

    if 4.7763004078433717e-164 < im < 2.51849397537875859e113

    1. Initial program 11.5

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

    if 2.51849397537875859e113 < im

    1. Initial program 53.4

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

    2. Taylor expanded around 0 4.9

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.776300407843372 \cdot 10^{-164}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 2.5184939753787586 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Reproduce

herbie shell --seed 2021207 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))