Average Error: 38.4 → 15.3
Time: 4.4s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.114175565259216 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.6621240162684264 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 5.926338806316416 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 9.367247253923697 \cdot 10^{+107} \lor \neg \left(re \leq 1.261312147033122 \cdot 10^{+149}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + \left(re \cdot 0.5\right) \cdot \frac{re}{im}\right) - re\right)}\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.114175565259216 \cdot 10^{+149}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -2.6621240162684264 \cdot 10^{-39}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \leq 5.926338806316416 \cdot 10^{+85}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \leq 9.367247253923697 \cdot 10^{+107} \lor \neg \left(re \leq 1.261312147033122 \cdot 10^{+149}\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 + \left(re \cdot 0.5\right) \cdot \frac{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 -6.114175565259216e+149)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -2.6621240162684264e-39)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 5.926338806316416e+85)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (if (or (<= re 9.367247253923697e+107)
               (not (<= re 1.261312147033122e+149)))
         (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 re))))
         (* 0.5 (sqrt (* 2.0 (- (+ im (* (* re 0.5) (/ 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 <= -6.114175565259216e+149) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -2.6621240162684264e-39) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 5.926338806316416e+85) {
		tmp = 0.5 * sqrt(2.0 * (im - re));
	} else if ((re <= 9.367247253923697e+107) || !(re <= 1.261312147033122e+149)) {
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt(1.0 / re));
	} else {
		tmp = 0.5 * sqrt(2.0 * ((im + ((re * 0.5) * (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 < -6.11417556525921569e149

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 7.7

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

    3. Simplified7.7

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

    if -6.11417556525921569e149 < re < -2.66212401626842645e-39

    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 *-un-lft-identity_binary6415.9

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

    if -2.66212401626842645e-39 < re < 5.9263388063164157e85

    1. Initial program 31.7

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

    2. Taylor expanded around 0 16.9

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

    if 5.9263388063164157e85 < re < 9.3672472539237e107 or 1.2613121470331221e149 < re

    1. Initial program 62.2

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

    2. Taylor expanded around 0 9.3

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

    3. Simplified9.3

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

    if 9.3672472539237e107 < re < 1.2613121470331221e149

    1. Initial program 53.5

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

    2. Taylor expanded around 0 44.4

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

    3. Simplified44.4

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

    5. Applied *-un-lft-identity_binary6444.4

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

    6. Applied times-frac_binary6444.4

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

    7. Applied associate-*r*_binary6444.4

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

    8. Simplified44.4

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.114175565259216 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.6621240162684264 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 5.926338806316416 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 9.367247253923697 \cdot 10^{+107} \lor \neg \left(re \leq 1.261312147033122 \cdot 10^{+149}\right):\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + \left(re \cdot 0.5\right) \cdot \frac{re}{im}\right) - re\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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)))))