Average Error: 32.1 → 18.2
Time: 1.3s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.030303640092379 \cdot 10^{+143}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.0769213379704504 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 8.533846604967322 \cdot 10^{-187}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 9.534116365977096 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.030303640092379 \cdot 10^{+143}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -1.0769213379704504 \cdot 10^{-254}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 8.533846604967322 \cdot 10^{-187}:\\
\;\;\;\;-im\\

\mathbf{elif}\;re \leq 9.534116365977096 \cdot 10^{+63}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.030303640092379e+143)
   (- re)
   (if (<= re -1.0769213379704504e-254)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 8.533846604967322e-187)
       (- im)
       (if (<= re 9.534116365977096e+63) (sqrt (+ (* re re) (* im im))) re)))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -1.030303640092379e+143) {
		tmp = -re;
	} else if (re <= -1.0769213379704504e-254) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= 8.533846604967322e-187) {
		tmp = -im;
	} else if (re <= 9.534116365977096e+63) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = 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 4 regimes

  2. if re < -1.03030364009237904e143

    1. Initial program 60.6

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

    2. Taylor expanded around -inf 8.2

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

    3. Simplified8.2

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

    if -1.03030364009237904e143 < re < -1.0769213379704504e-254 or 8.53384660496732172e-187 < re < 9.5341163659770961e63

    1. Initial program 19.0

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

    if -1.0769213379704504e-254 < re < 8.53384660496732172e-187

    1. Initial program 31.1

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

    2. Taylor expanded around -inf 32.9

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

    3. Simplified32.9

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

    if 9.5341163659770961e63 < re

    1. Initial program 47.3

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

    2. Taylor expanded around inf 12.4

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.030303640092379 \cdot 10^{+143}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.0769213379704504 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 8.533846604967322 \cdot 10^{-187}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 9.534116365977096 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2021009 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))