Average Error: 38.0 → 11.5
Time: 3.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 4.41283550356444898 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 4.41283550356444898 \cdot 10^{151}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r17301 = 0.5;
        double r17302 = 2.0;
        double r17303 = re;
        double r17304 = r17303 * r17303;
        double r17305 = im;
        double r17306 = r17305 * r17305;
        double r17307 = r17304 + r17306;
        double r17308 = sqrt(r17307);
        double r17309 = r17308 - r17303;
        double r17310 = r17302 * r17309;
        double r17311 = sqrt(r17310);
        double r17312 = r17301 * r17311;
        return r17312;
}
↓
double f(double re, double im) {
        double r17313 = re;
        double r17314 = 4.412835503564449e+151;
        bool r17315 = r17313 <= r17314;
        double r17316 = 0.5;
        double r17317 = 2.0;
        double r17318 = 1.0;
        double r17319 = im;
        double r17320 = hypot(r17313, r17319);
        double r17321 = r17318 * r17320;
        double r17322 = r17321 - r17313;
        double r17323 = r17317 * r17322;
        double r17324 = sqrt(r17323);
        double r17325 = r17316 * r17324;
        double r17326 = 2.0;
        double r17327 = pow(r17319, r17326);
        double r17328 = 0.0;
        double r17329 = r17327 + r17328;
        double r17330 = r17313 + r17320;
        double r17331 = r17329 / r17330;
        double r17332 = r17317 * r17331;
        double r17333 = sqrt(r17332);
        double r17334 = r17316 * r17333;
        double r17335 = r17315 ? r17325 : r17334;
        return r17335;
}

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 < 4.412835503564449e+151

    1. Initial program 34.2

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

    3. Applied *-un-lft-identity34.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}} - re\right)}\]

    4. Applied sqrt-prod34.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}} - re\right)}\]

    5. Simplified34.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\]

    6. Simplified8.7

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

    if 4.412835503564449e+151 < re

    1. Initial program 63.9

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

    3. Applied flip--63.9

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

    4. Simplified50.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified30.4

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 4.41283550356444898 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))))