Average Error: 38.6 → 19.6
Time: 3.2s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.832367929892014 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.3334948043090795 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.4182296952350283 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 4.0290642257397116 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.832367929892014 \cdot 10^{+108}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -2.3334948043090795 \cdot 10^{-16}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \leq -1.4182296952350283 \cdot 10^{-136}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \leq 4.0290642257397116 \cdot 10^{+68}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{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 -6.832367929892014e+108)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -2.3334948043090795e-16)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re -1.4182296952350283e-136)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
       (if (<= re 4.0290642257397116e+68)
         (* 0.5 (sqrt (* 2.0 im)))
         (* 0.5 (sqrt (* 2.0 (* 0.5 (/ (pow im 2.0) 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 <= -6.832367929892014e+108) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -2.3334948043090795e-16) {
		tmp = 0.5 * sqrt(2.0 * (im - re));
	} else if (re <= -1.4182296952350283e-136) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 4.0290642257397116e+68) {
		tmp = 0.5 * sqrt(2.0 * im);
	} else {
		tmp = 0.5 * sqrt(2.0 * (0.5 * (pow(im, 2.0) / 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 5 regimes

  2. if re < -6.8323679298920138e108

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 9.7

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

    if -6.8323679298920138e108 < re < -2.33349480430907953e-16

    1. Initial program 16.8

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

    2. Taylor expanded around 0 31.0

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

    if -2.33349480430907953e-16 < re < -1.41822969523502832e-136

    1. Initial program 15.5

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

    if -1.41822969523502832e-136 < re < 4.0290642257397116e68

    1. Initial program 33.6

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

    2. Taylor expanded around 0 15.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}}\]

    if 4.0290642257397116e68 < re

    1. Initial program 59.5

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

    2. Taylor expanded around inf 32.7

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

  4. Final simplification19.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.832367929892014 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.3334948043090795 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.4182296952350283 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 4.0290642257397116 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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)))))