Average Error: 38.1 → 11.5
Time: 4.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164} \lor \neg \left(re \le -1.2762910995619203 \cdot 10^{125} \lor \neg \left(re \le -9.25977116271673673 \cdot 10^{-39}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164} \lor \neg \left(re \le -1.2762910995619203 \cdot 10^{125} \lor \neg \left(re \le -9.25977116271673673 \cdot 10^{-39}\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(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\

\end{array}
double f(double re, double im) {
        double r167507 = 0.5;
        double r167508 = 2.0;
        double r167509 = re;
        double r167510 = r167509 * r167509;
        double r167511 = im;
        double r167512 = r167511 * r167511;
        double r167513 = r167510 + r167512;
        double r167514 = sqrt(r167513);
        double r167515 = r167514 + r167509;
        double r167516 = r167508 * r167515;
        double r167517 = sqrt(r167516);
        double r167518 = r167507 * r167517;
        return r167518;
}
↓
double f(double re, double im) {
        double r167519 = re;
        double r167520 = -5.840879447248876e+164;
        bool r167521 = r167519 <= r167520;
        double r167522 = -1.2762910995619203e+125;
        bool r167523 = r167519 <= r167522;
        double r167524 = -9.259771162716737e-39;
        bool r167525 = r167519 <= r167524;
        double r167526 = !r167525;
        bool r167527 = r167523 || r167526;
        double r167528 = !r167527;
        bool r167529 = r167521 || r167528;
        double r167530 = 0.5;
        double r167531 = 2.0;
        double r167532 = im;
        double r167533 = r167532 * r167532;
        double r167534 = hypot(r167519, r167532);
        double r167535 = r167534 - r167519;
        double r167536 = r167533 / r167535;
        double r167537 = r167531 * r167536;
        double r167538 = sqrt(r167537);
        double r167539 = r167530 * r167538;
        double r167540 = 1.0;
        double r167541 = r167519 + r167534;
        double r167542 = r167540 * r167541;
        double r167543 = r167531 * r167542;
        double r167544 = sqrt(r167543);
        double r167545 = r167530 * r167544;
        double r167546 = r167529 ? r167539 : r167545;
        return r167546;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie11.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -5.840879447248876e+164 or -1.2762910995619203e+125 < re < -9.259771162716737e-39

    1. Initial program 55.4

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

    3. Applied flip-+55.4

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

    4. Simplified40.1

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

    5. Simplified30.5

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

    if -5.840879447248876e+164 < re < -1.2762910995619203e+125 or -9.259771162716737e-39 < re

    1. Initial program 32.5

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

    3. Applied *-un-lft-identity32.5

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

    4. Applied *-un-lft-identity32.5

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

    5. Applied distribute-lft-out32.5

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

    6. Simplified5.4

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164} \lor \neg \left(re \le -1.2762910995619203 \cdot 10^{125} \lor \neg \left(re \le -9.25977116271673673 \cdot 10^{-39}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))))