Average Error: 31.2 → 7.3
Time: 3.5s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.7917667196950207 \cdot 10^{+107}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -8.034323118303542 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.7917667196950207 \cdot 10^{+107}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -8.034323118303542 \cdot 10^{-165}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.7917667196950207e+107)
   (- re)
   (if (<= re -8.034323118303542e-165) (sqrt (+ (* re re) (* im im))) im)))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -1.7917667196950207e+107) {
		tmp = -re;
	} else if (re <= -8.034323118303542e-165) {
		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 re < -1.79176671969502073e107

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 6.0

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

    if -1.79176671969502073e107 < re < -8.03432311830354215e-165

    1. Initial program 10.9

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

    if -8.03432311830354215e-165 < re

    1. Initial program 32.0

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

    2. Taylor expanded around 0 5.4

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7917667196950207 \cdot 10^{+107}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -8.034323118303542 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))