Average Error: 38.7 → 11.2
Time: 4.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -374104356870144197000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -374104356870144197000:\\
\;\;\;\;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(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r288346 = 0.5;
        double r288347 = 2.0;
        double r288348 = re;
        double r288349 = r288348 * r288348;
        double r288350 = im;
        double r288351 = r288350 * r288350;
        double r288352 = r288349 + r288351;
        double r288353 = sqrt(r288352);
        double r288354 = r288353 + r288348;
        double r288355 = r288347 * r288354;
        double r288356 = sqrt(r288355);
        double r288357 = r288346 * r288356;
        return r288357;
}
↓
double f(double re, double im) {
        double r288358 = re;
        double r288359 = -3.741043568701442e+20;
        bool r288360 = r288358 <= r288359;
        double r288361 = 0.5;
        double r288362 = 2.0;
        double r288363 = im;
        double r288364 = r288363 * r288363;
        double r288365 = hypot(r288358, r288363);
        double r288366 = r288365 - r288358;
        double r288367 = r288364 / r288366;
        double r288368 = r288362 * r288367;
        double r288369 = sqrt(r288368);
        double r288370 = r288361 * r288369;
        double r288371 = 1.0;
        double r288372 = r288371 * r288365;
        double r288373 = r288372 + r288358;
        double r288374 = r288362 * r288373;
        double r288375 = sqrt(r288374);
        double r288376 = r288361 * r288375;
        double r288377 = r288360 ? r288370 : r288376;
        return r288377;
}

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
Target33.5
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -3.741043568701442e+20

    1. Initial program 58.4

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

    3. Applied flip-+58.4

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

    4. Simplified41.7

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

    5. Simplified30.1

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

    if -3.741043568701442e+20 < re

    1. Initial program 32.7

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

    3. Applied *-un-lft-identity32.7

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

    4. Applied sqrt-prod32.7

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

    5. Simplified32.7

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

    6. Simplified5.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -374104356870144197000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))))