Average Error: 38.8 → 15.2
Time: 4.8s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -4.020338852037762 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.0383523502862224 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.318656067702442 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 9.463404869897915 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -4.020338852037762 \cdot 10^{+126}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -1.0383523502862224 \cdot 10^{+81}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \leq -2.318656067702442 \cdot 10^{-150}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \leq 9.463404869897915 \cdot 10^{+67}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\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 -4.020338852037762e+126)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -1.0383523502862224e+81)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re -2.318656067702442e-150)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
       (if (<= re 9.463404869897915e+67)
         (* 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 <= -4.020338852037762e+126) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -1.0383523502862224e+81) {
		tmp = 0.5 * sqrt(2.0 * (im - re));
	} else if (re <= -2.318656067702442e-150) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 9.463404869897915e+67) {
		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 5 regimes

  2. if re < -4.02033885203776216e126

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 9.1

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

    3. Simplified9.1

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

    if -4.02033885203776216e126 < re < -1.03835235028622243e81

    1. Initial program 21.3

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

    2. Taylor expanded around 0 32.1

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

    if -1.03835235028622243e81 < re < -2.3186560677024421e-150

    1. Initial program 17.0

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

    3. Applied add-cube-cbrt_binary6417.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - \color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}\right)} \]

    4. Applied cancel-sign-sub-inv_binary6417.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}} \]

    5. Simplified17.0

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

    if -2.3186560677024421e-150 < re < 9.4634048698979148e67

    1. Initial program 34.3

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

    2. Taylor expanded around 0 16.2

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

    3. Simplified16.2

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

    if 9.4634048698979148e67 < re

    1. Initial program 60.1

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

    2. Taylor expanded around 0 12.3

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

    3. Simplified12.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.020338852037762 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.0383523502862224 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.318656067702442 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 9.463404869897915 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021197 
(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)))))