Average Error: 38.7 → 6.3
Time: 3.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.6699074942752878 \cdot 10^{-177}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.6699074942752878 \cdot 10^{-177}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\

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

\end{array}
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 VAR;
	if ((re <= -7.669907494275288e-177)) {
		VAR = (0.5 * sqrt((2.0 * (im / ((hypot(re, im) - re) / im)))));
	} else {
		VAR = (0.5 * sqrt((2.0 * (hypot(im, re) + re))));
	}
	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

Target

Original38.7
Target34.1
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -7.669907494275288e-177

    1. Initial program 50.4

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

    3. Applied flip-+50.3

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

    4. Simplified38.9

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

    5. Simplified32.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]
    6. Using strategy rm

    7. Applied unpow232.6

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

    8. Applied associate-/l*14.2

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

    if -7.669907494275288e-177 < re

    1. Initial program 31.1

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

    3. Applied +-commutative31.1

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

    4. Applied hypot-def1.2

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.6699074942752878 \cdot 10^{-177}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))