Average Error: 37.7 → 11.3
Time: 6.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.93226235881665375 \cdot 10^{87} \lor \neg \left(re \le -4.352408394580921 \cdot 10^{21} \lor \neg \left(re \le -2.8486185678520585 \cdot 10^{-26}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.93226235881665375 \cdot 10^{87} \lor \neg \left(re \le -4.352408394580921 \cdot 10^{21} \lor \neg \left(re \le -2.8486185678520585 \cdot 10^{-26}\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\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 <= -8.932262358816654e+87) || !((re <= -4.352408394580921e+21) || !(re <= -2.8486185678520585e-26)))) {
		VAR = (0.5 * sqrt((2.0 * ((im * im) / (hypot(re, im) - re)))));
	} else {
		VAR = (0.5 * sqrt((2.0 * (hypot(re, im) + 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

Original37.7
Target32.3
Herbie11.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -8.932262358816654e+87 or -4.352408394580921e+21 < re < -2.8486185678520585e-26

    1. Initial program 57.6

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

    3. Applied flip-+57.6

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

    4. Simplified40.9

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

    5. Simplified30.3

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

    if -8.932262358816654e+87 < re < -4.352408394580921e+21 or -2.8486185678520585e-26 < re

    1. Initial program 32.3

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

    3. Applied hypot-def6.2

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.93226235881665375 \cdot 10^{87} \lor \neg \left(re \le -4.352408394580921 \cdot 10^{21} \lor \neg \left(re \le -2.8486185678520585 \cdot 10^{-26}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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)))))