Average Error: 31.1 → 17.2
Time: 1.7s
Precision: binary64
Cost: 8132
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -4.1162129695124836 \cdot 10^{+151}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq -2.1964806768619867 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \leq 2.11287353661743 \cdot 10^{-238}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 7.549554504255097 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq -4.1162129695124836 \cdot 10^{+151}:\\
\;\;\;\;-im\\

\mathbf{elif}\;im \leq -2.1964806768619867 \cdot 10^{-223}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;im \leq 2.11287353661743 \cdot 10^{-238}:\\
\;\;\;\;-re\\

\mathbf{elif}\;im \leq 7.549554504255097 \cdot 10^{+118}:\\
\;\;\;\;\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 -4.1162129695124836e+151)
   (- im)
   (if (<= im -2.1964806768619867e-223)
     (sqrt (+ (* re re) (* im im)))
     (if (<= im 2.11287353661743e-238)
       (- re)
       (if (<= im 7.549554504255097e+118)
         (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 <= -4.1162129695124836e+151) {
		tmp = -im;
	} else if (im <= -2.1964806768619867e-223) {
		tmp = sqrt((re * re) + (im * im));
	} else if (im <= 2.11287353661743e-238) {
		tmp = -re;
	} else if (im <= 7.549554504255097e+118) {
		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

Alternatives

Alternative 1
Error25.7
Cost1027
\[\begin{array}{l} \mathbf{if}\;im \leq -2.8415552706869677 \cdot 10^{-93}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 8.015064203069834 \cdot 10^{-242}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 210377114512.3621:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
Alternative 2
Error26.5
Cost706
\[\begin{array}{l} \mathbf{if}\;im \leq -3.4908880737490054 \cdot 10^{-151}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4099513107437.5083:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
Alternative 3
Error36.8
Cost1027
\[\begin{array}{l} \mathbf{if}\;re \leq 3.303349602246751 \cdot 10^{-148}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5.865007045303084 \cdot 10^{-68}:\\ \;\;\;\;re\\ \mathbf{elif}\;re \leq 5.532214216476038 \cdot 10^{-37}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
Alternative 4
Error46.0
Cost385
\[\begin{array}{l} \mathbf{if}\;re \leq -3.007955690093281 \cdot 10^{-307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
Alternative 5
Error60.5
Cost64
\[1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if im < -4.11621296951248365e151

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 7.0

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

    3. Simplified7.0

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

    4. Simplified7.0

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

    if -4.11621296951248365e151 < im < -2.1964806768619867e-223 or 2.1128735366174298e-238 < im < 7.549554504255097e118

    1. Initial program 18.1

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

    2. Simplified18.1

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

    if -2.1964806768619867e-223 < im < 2.1128735366174298e-238

    1. Initial program 30.4

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

    2. Taylor expanded around -inf 32.4

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

    3. Simplified32.4

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

    4. Simplified32.4

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

    if 7.549554504255097e118 < im

    1. Initial program 54.8

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

    2. Taylor expanded around 0 9.5

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

    3. Simplified9.5

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1162129695124836 \cdot 10^{+151}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq -2.1964806768619867 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \leq 2.11287353661743 \cdot 10^{-238}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 7.549554504255097 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))