Average Error: 38.3 → 13.3
Time: 4.4s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.267084959064262 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -8.516731262603456 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 1.9790760428740222 \cdot 10^{-07}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.267084959064262 \cdot 10^{+99}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -8.516731262603456 \cdot 10^{-164}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\

\mathbf{elif}\;re \leq 1.9790760428740222 \cdot 10^{-07}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{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 -1.267084959064262e+99)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -8.516731262603456e-164)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* im im) (* re re))) re))))
     (if (<= re 1.9790760428740222e-07)
       (* 0.5 (sqrt (* 2.0 (- (+ im (* 0.5 (* re (/ re im)))) re))))
       (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 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 <= -1.267084959064262e+99) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -8.516731262603456e-164) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((im * im) + (re * re)) - re));
	} else if (re <= 1.9790760428740222e-07) {
		tmp = 0.5 * sqrt(2.0 * ((im + (0.5 * (re * (re / im)))) - re));
	} else {
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt(1.0 / 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 < -1.26708495906426203e99

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 9.7

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

    3. Simplified9.7

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

    if -1.26708495906426203e99 < re < -8.5167312626034563e-164

    1. Initial program 15.9

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

    3. Applied +-commutative_binary6415.9

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

    if -8.5167312626034563e-164 < re < 1.97907604287402217e-7

    1. Initial program 32.5

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

    2. Taylor expanded around 0 12.3

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

    3. Simplified12.3

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

    5. Applied *-un-lft-identity_binary6412.3

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

    6. Applied times-frac_binary6412.4

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

    7. Simplified12.4

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

    if 1.97907604287402217e-7 < re

    1. Initial program 56.6

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

    2. Taylor expanded around 0 14.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.267084959064262 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -8.516731262603456 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \leq 1.9790760428740222 \cdot 10^{-07}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(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)))))