Average Error: 38.2 → 13.6
Time: 5.4s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.6442239073885936 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.9511813760720313 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 5.235325772799965 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\ \mathbf{elif}\;re \leq 86846314.06309874 \lor \neg \left(re \leq 2.588079697153549 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.6442239073885936 \cdot 10^{+151}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -1.9511813760720313 \cdot 10^{-122}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \leq 5.235325772799965 \cdot 10^{-30}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\

\mathbf{elif}\;re \leq 86846314.06309874 \lor \neg \left(re \leq 2.588079697153549 \cdot 10^{+33}\right):\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\


\end{array}

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -2.6442239073885936e+151)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -1.9511813760720313e-122)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 5.235325772799965e-30)
       (* 0.5 (sqrt (* 2.0 (- (+ im (* 0.5 (* re (/ re im)))) re))))
       (if (or (<= re 86846314.06309874) (not (<= re 2.588079697153549e+33)))
         (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 re))))
         (* 0.5 (sqrt (* 2.0 (- (+ im (* 0.5 (/ (* re re) im))) re)))))))))
double code(double re, double im) {
	return 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -2.6442239073885936e+151) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -1.9511813760720313e-122) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 5.235325772799965e-30) {
		tmp = 0.5 * sqrt(2.0 * ((im + (0.5 * (re * (re / im)))) - re));
	} else if ((re <= 86846314.06309874) || !(re <= 2.588079697153549e+33)) {
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt(1.0 / re));
	} else {
		tmp = 0.5 * sqrt(2.0 * ((im + (0.5 * ((re * re) / im))) - 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 5 regimes

  2. if re < -2.6442239073885936e151

    1. Initial program 62.5

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Taylor expanded around -inf 8.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]

    3. Simplified8.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

    if -2.6442239073885936e151 < re < -1.95118137607203126e-122

    1. Initial program 15.5

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary6415.5

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

    if -1.95118137607203126e-122 < re < 5.2353257727999647e-30

    1. Initial program 29.9

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Taylor expanded around 0 12.0

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

    3. Simplified12.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)} - re\right)} \]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary6412.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{\color{blue}{1 \cdot im}}\right) - re\right)} \]

    6. Applied times-frac_binary6412.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \color{blue}{\left(\frac{re}{1} \cdot \frac{re}{im}\right)}\right) - re\right)} \]

    7. Simplified12.0

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

    if 5.2353257727999647e-30 < re < 86846314.0630987436 or 2.58807969715354886e33 < re

    1. Initial program 56.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Taylor expanded around 0 15.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

    if 86846314.0630987436 < re < 2.58807969715354886e33

    1. Initial program 48.8

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Taylor expanded around 0 31.3

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

    3. Simplified31.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)} - re\right)} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.6442239073885936 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.9511813760720313 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 5.235325772799965 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\ \mathbf{elif}\;re \leq 86846314.06309874 \lor \neg \left(re \leq 2.588079697153549 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021196 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))