Average Error: 38.1 → 11.3
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 4.717568369814920763211379241631962287037 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 4.717568369814920763211379241631962287037 \cdot 10^{-15}:\\
\;\;\;\;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 r16383 = 0.5;
        double r16384 = 2.0;
        double r16385 = re;
        double r16386 = r16385 * r16385;
        double r16387 = im;
        double r16388 = r16387 * r16387;
        double r16389 = r16386 + r16388;
        double r16390 = sqrt(r16389);
        double r16391 = r16390 - r16385;
        double r16392 = r16384 * r16391;
        double r16393 = sqrt(r16392);
        double r16394 = r16383 * r16393;
        return r16394;
}
↓
double f(double re, double im) {
        double r16395 = re;
        double r16396 = 4.717568369814921e-15;
        bool r16397 = r16395 <= r16396;
        double r16398 = 0.5;
        double r16399 = 2.0;
        double r16400 = 1.0;
        double r16401 = im;
        double r16402 = hypot(r16395, r16401);
        double r16403 = r16400 * r16402;
        double r16404 = r16403 - r16395;
        double r16405 = r16399 * r16404;
        double r16406 = sqrt(r16405);
        double r16407 = r16398 * r16406;
        double r16408 = 2.0;
        double r16409 = pow(r16401, r16408);
        double r16410 = 0.0;
        double r16411 = r16409 + r16410;
        double r16412 = r16395 + r16402;
        double r16413 = r16411 / r16412;
        double r16414 = r16399 * r16413;
        double r16415 = sqrt(r16414);
        double r16416 = r16398 * r16415;
        double r16417 = r16397 ? r16407 : r16416;
        return r16417;
}

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 < 4.717568369814921e-15

    1. Initial program 31.9

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

    3. Applied *-un-lft-identity31.9

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

    4. Applied sqrt-prod31.9

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

    5. Simplified31.9

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

    6. Simplified4.6

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

    if 4.717568369814921e-15 < re

    1. Initial program 55.1

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

    3. Applied flip--55.1

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

    4. Simplified38.9

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

    5. Simplified29.9

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 4.717568369814920763211379241631962287037 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +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)))))