Average Error: 38.5 → 14.0
Time: 8.7s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.288187725469977 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -4.634337825695654 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 8.321275884393451 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.288187725469977 \cdot 10^{+32}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\

\mathbf{elif}\;re \leq -4.634337825695654 \cdot 10^{-125}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \leq 8.321275884393451 \cdot 10^{-24}:\\
\;\;\;\;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 -9.288187725469977e+32)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re -4.634337825695654e-125)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))))
     (if (<= re 8.321275884393451e-24)
       (* 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 <= -9.288187725469977e+32) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -4.634337825695654e-125) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 8.321275884393451e-24) {
		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 < -9.28818772546997719e32

    1. Initial program 41.7

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

    2. Taylor expanded around -inf 13.2

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

    3. Simplified13.2

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

    if -9.28818772546997719e32 < re < -4.63433782569565382e-125

    1. Initial program 16.0

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

    if -4.63433782569565382e-125 < re < 8.321275884393451e-24

    1. Initial program 31.6

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

    2. Taylor expanded around 0 11.7

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

    3. Simplified11.7

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

    if 8.321275884393451e-24 < re

    1. Initial program 56.1

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

    2. Taylor expanded around 0 16.9

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

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.288187725469977 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -4.634337825695654 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 8.321275884393451 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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)))))