Average Error: 38.5 → 12.1
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.31302022362198529 \cdot 10^{-40} \lor \neg \left(re \le 3.84256221587197863 \cdot 10^{121} \lor \neg \left(re \le 1.00281996477980382 \cdot 10^{194}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.31302022362198529 \cdot 10^{-40} \lor \neg \left(re \le 3.84256221587197863 \cdot 10^{121} \lor \neg \left(re \le 1.00281996477980382 \cdot 10^{194}\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
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 VAR;
	if (((re <= 1.3130202236219853e-40) || !((re <= 3.8425622158719786e+121) || !(re <= 1.0028199647798038e+194)))) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) hypot(re, im)) - re))))))));
	} else {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) fma(im, im, 0.0)) / ((double) (re + ((double) hypot(re, im))))))))))));
	}
	return VAR;
}

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 2 regimes

  2. if re < 1.3130202236219853e-40 or 3.8425622158719786e+121 < re < 1.0028199647798038e+194

    1. Initial program 34.1

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

    3. Applied hypot-def6.7

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

    if 1.3130202236219853e-40 < re < 3.8425622158719786e+121 or 1.0028199647798038e+194 < re

    1. Initial program 54.4

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

    3. Applied flip--54.4

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

    4. Simplified39.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\mathsf{fma}\left(im, im, 0\right)}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified31.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.31302022362198529 \cdot 10^{-40} \lor \neg \left(re \le 3.84256221587197863 \cdot 10^{121} \lor \neg \left(re \le 1.00281996477980382 \cdot 10^{194}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))