Average Error: 32.0 → 7.6
Time: 1.8s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 8.285117424768534 \cdot 10^{-162}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 1.146743099784204 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im + 0.5 \cdot \frac{re}{\frac{im}{re}}\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 8.285117424768534 \cdot 10^{-162}:\\
\;\;\;\;-re\\

\mathbf{elif}\;im \leq 1.146743099784204 \cdot 10^{+114}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;im + 0.5 \cdot \frac{re}{\frac{im}{re}}\\

\end{array}
(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore modulus (re im)
 :precision binary64
 (if (<= im 8.285117424768534e-162)
   (- re)
   (if (<= im 1.146743099784204e+114)
     (sqrt (+ (* re re) (* im im)))
     (+ im (* 0.5 (/ re (/ im re)))))))
double modulus(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double modulus(double re, double im) {
	double tmp;
	if (im <= 8.285117424768534e-162) {
		tmp = -re;
	} else if (im <= 1.146743099784204e+114) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = 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 im < 8.2851174247685339e-162

    1. Initial program 33.4

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

    2. Taylor expanded around -inf 5.7

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

    3. Simplified5.7

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

    if 8.2851174247685339e-162 < im < 1.146743099784204e114

    1. Initial program 11.1

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

    if 1.146743099784204e114 < im

    1. Initial program 53.8

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

    2. Taylor expanded around 0 13.2

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

    3. Simplified13.2

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

    5. Applied associate-/l*_binary646.1

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.285117424768534 \cdot 10^{-162}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 1.146743099784204 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im + 0.5 \cdot \frac{re}{\frac{im}{re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))