Average Error: 31.3 → 7.3
Time: 1.1s
Precision: binary64
[re im]: =sort([re im])
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 2.715301714894546 \cdot 10^{-140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 1.0282379563465237 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 2.715301714894546 \cdot 10^{-140}:\\
\;\;\;\;-re\\

\mathbf{elif}\;im \leq 1.0282379563465237 \cdot 10^{+127}:\\
\;\;\;\;\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 (<= im 2.715301714894546e-140)
   (- re)
   (if (<= im 1.0282379563465237e+127) (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 (im <= 2.715301714894546e-140) {
		tmp = -re;
	} else if (im <= 1.0282379563465237e+127) {
		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 < 2.7153017148945459e-140

    1. Initial program 31.6

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

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

    if 2.7153017148945459e-140 < im < 1.02823795634652372e127

    1. Initial program 10.7

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

    if 1.02823795634652372e127 < im

    1. Initial program 55.4

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

    2. Taylor expanded around 0 4.9

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.715301714894546 \cdot 10^{-140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 1.0282379563465237 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2020358 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))