Average Error: 38.2 → 21.1
Time: 4.6s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.6385368300839155 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -5.686701427801653 \cdot 10^{-162}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re\right)}\\ \mathbf{elif}\;re \leq 3.6532567719110757 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.301551743401769 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \leq 8.594773102387256 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.6385368300839155 \cdot 10^{+81}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -5.686701427801653 \cdot 10^{-162}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re\right)}\\

\mathbf{elif}\;re \leq 3.6532567719110757 \cdot 10^{-75}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \leq 2.301551743401769 \cdot 10^{+110}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\

\mathbf{elif}\;re \leq 8.594773102387256 \cdot 10^{+176}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot 0\\

\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 -1.6385368300839155e+81)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -5.686701427801653e-162)
     (* 0.5 (sqrt (* 2.0 (- (exp (log (sqrt (+ (* re re) (* im im))))) re))))
     (if (<= re 3.6532567719110757e-75)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (if (<= re 2.301551743401769e+110)
         (*
          0.5
          (/
           (sqrt (* 2.0 (* im im)))
           (sqrt (+ re (sqrt (+ (* re re) (* im im)))))))
         (if (<= re 8.594773102387256e+176)
           (* 0.5 (sqrt (* 2.0 (- im re))))
           (* 0.5 0.0)))))))
double code(double re, double im) {
	return ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
}
↓
double code(double re, double im) {
	double tmp;
	if ((re <= -1.6385368300839155e+81)) {
		tmp = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re * -2.0))))))));
	} else {
		double tmp_1;
		if ((re <= -5.686701427801653e-162)) {
			tmp_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) exp(((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) - re))))))));
		} else {
			double tmp_2;
			if ((re <= 3.6532567719110757e-75)) {
				tmp_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
			} else {
				double tmp_3;
				if ((re <= 2.301551743401769e+110)) {
					tmp_3 = ((double) (0.5 * (((double) sqrt(((double) (2.0 * ((double) (im * im)))))) / ((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))));
				} else {
					double tmp_4;
					if ((re <= 8.594773102387256e+176)) {
						tmp_4 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
					} else {
						tmp_4 = ((double) (0.5 * 0.0));
					}
					tmp_3 = tmp_4;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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 < -1.6385368300839155e81

    1. Initial program Error: 48.7 bits

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

    2. Taylor expanded around -inf Error: 10.7 bits

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

    3. SimplifiedError: 10.7 bits

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

    if -1.6385368300839155e81 < re < -5.68670142780165302e-162

    1. Initial program Error: 15.2 bits

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

    3. Applied add-exp-logError: 17.7 bits

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

    if -5.68670142780165302e-162 < re < 3.65325677191107568e-75 or 2.30155174340176889e110 < re < 8.59477310238725624e176

    1. Initial program Error: 34.4 bits

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

    2. Taylor expanded around 0 Error: 15.7 bits

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

    if 3.65325677191107568e-75 < re < 2.30155174340176889e110

    1. Initial program Error: 46.2 bits

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

    3. Applied flip--Error: 46.2 bits

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

    4. Applied associate-*r/Error: 46.3 bits

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

    5. Applied sqrt-divError: 46.4 bits

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

    6. SimplifiedError: 31.0 bits

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

    7. SimplifiedError: 31.0 bits

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

    if 8.59477310238725624e176 < re

    1. Initial program Error: 64.0 bits

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

    2. Taylor expanded around inf Error: 50.1 bits

      \[\leadsto 0.5 \cdot \color{blue}{0}\]
  3. Recombined 5 regimes into one program.

  4. Final simplificationError: 21.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6385368300839155 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -5.686701427801653 \cdot 10^{-162}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re\right)}\\ \mathbf{elif}\;re \leq 3.6532567719110757 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.301551743401769 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \leq 8.594773102387256 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))))