Average Error: 39.1 → 12.3
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.8460888780624261 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.8460888780624261 \cdot 10^{-106}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\

\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 ((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 <= -2.846088878062426e-106)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (im * im)) / ((double) fma(-1.0, re, ((double) hypot(re, im))))))))))));
	} else {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) 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

Original39.1
Target34.1
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -2.846088878062426e-106

    1. Initial program 53.2

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

    3. Applied flip-+53.2

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

    4. Simplified39.0

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

    5. Simplified31.6

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

    if -2.846088878062426e-106 < re

    1. Initial program 31.8

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

    3. Applied hypot-def2.4

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.8460888780624261 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))))