Average Error: 31.4 → 19.0
Time: 2.0s
Precision: binary64
Cost: 8774
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -3.148243747314542 \cdot 10^{+78}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq -6.207988187895342 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;im \leq -2.4637364856279063 \cdot 10^{-252}:\\ \;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\ \mathbf{elif}\;im \leq 4.045551410041801 \cdot 10^{-272}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.325491675883215 \cdot 10^{-188}:\\ \;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\ \mathbf{elif}\;im \leq 2.4191264962337645 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq -3.148243747314542 \cdot 10^{+78}:\\
\;\;\;\;-im\\

\mathbf{elif}\;im \leq -6.207988187895342 \cdot 10^{-190}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;im \leq -2.4637364856279063 \cdot 10^{-252}:\\
\;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\

\mathbf{elif}\;im \leq 4.045551410041801 \cdot 10^{-272}:\\
\;\;\;\;re\\

\mathbf{elif}\;im \leq 1.325491675883215 \cdot 10^{-188}:\\
\;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\

\mathbf{elif}\;im \leq 2.4191264962337645 \cdot 10^{+94}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im -3.148243747314542e+78)
   (- im)
   (if (<= im -6.207988187895342e-190)
     (sqrt (+ (* im im) (* re re)))
     (if (<= im -2.4637364856279063e-252)
       (- (* (/ (* im im) re) -0.5) re)
       (if (<= im 4.045551410041801e-272)
         re
         (if (<= im 1.325491675883215e-188)
           (- (* (/ (* im im) re) -0.5) re)
           (if (<= im 2.4191264962337645e+94)
             (sqrt (+ (* im im) (* re re)))
             im)))))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= -3.148243747314542e+78) {
		tmp = -im;
	} else if (im <= -6.207988187895342e-190) {
		tmp = sqrt((im * im) + (re * re));
	} else if (im <= -2.4637364856279063e-252) {
		tmp = (((im * im) / re) * -0.5) - re;
	} else if (im <= 4.045551410041801e-272) {
		tmp = re;
	} else if (im <= 1.325491675883215e-188) {
		tmp = (((im * im) / re) * -0.5) - re;
	} else if (im <= 2.4191264962337645e+94) {
		tmp = sqrt((im * im) + (re * re));
	} 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

Alternatives

Alternative 1
Error33.8
Cost19648
\[e^{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}\]
Alternative 2
Error31.4
Cost6848
\[\sqrt{im \cdot im + re \cdot re}\]
Alternative 3
Error48.3
Cost576
\[\frac{im \cdot im}{re} \cdot -0.5 - re\]
Alternative 4
Error47.1
Cost128
\[-im\]
Alternative 5
Error46.8
Cost128
\[-re\]
Alternative 6
Error46.8
Cost64
\[re\]
Alternative 7
Error46.7
Cost64
\[im\]
Alternative 8
Error60.6
Cost64
\[1\]
Alternative 9
Error62.3
Cost64
\[0\]
Alternative 10
Error63.2
Cost64
\[-1\]

Error

Derivation

  1. Split input into 5 regimes

  2. if im < -3.14824374731454187e78

    1. Initial program 48.2

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

    2. Taylor expanded around -inf 12.2

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

    3. Simplified12.2

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

    4. Simplified12.2

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

    if -3.14824374731454187e78 < im < -6.2079881878953418e-190 or 1.32549167588321498e-188 < im < 2.4191264962337645e94

    1. Initial program 18.0

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

    2. Simplified18.0

      \[\leadsto \color{blue}{\sqrt{im \cdot im + re \cdot re}}\]

    if -6.2079881878953418e-190 < im < -2.4637364856279063e-252 or 4.0455514100418009e-272 < im < 1.32549167588321498e-188

    1. Initial program 28.9

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

    2. Taylor expanded around -inf 36.4

      \[\leadsto \color{blue}{-\left(re + 0.5 \cdot \frac{{im}^{2}}{re}\right)}\]

    3. Simplified36.4

      \[\leadsto \color{blue}{\frac{im \cdot im}{re} \cdot -0.5 - re}\]

    4. Simplified36.4

      \[\leadsto \color{blue}{\frac{im \cdot im}{re} \cdot -0.5 - re}\]

    if -2.4637364856279063e-252 < im < 4.0455514100418009e-272

    1. Initial program 29.6

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

    2. Taylor expanded around inf 36.4

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

    3. Simplified36.4

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

    if 2.4191264962337645e94 < im

    1. Initial program 49.8

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

    2. Taylor expanded around 0 9.6

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

    3. Simplified9.6

      \[\leadsto \color{blue}{im}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.148243747314542 \cdot 10^{+78}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq -6.207988187895342 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;im \leq -2.4637364856279063 \cdot 10^{-252}:\\ \;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\ \mathbf{elif}\;im \leq 4.045551410041801 \cdot 10^{-272}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.325491675883215 \cdot 10^{-188}:\\ \;\;\;\;\frac{im \cdot im}{re} \cdot -0.5 - re\\ \mathbf{elif}\;im \leq 2.4191264962337645 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))