Average Error: 38.6 → 26.4
Time: 8.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{{re}^{2} + {im}^{2}} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\
\;\;\;\;\sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{{re}^{2} + {im}^{2}} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\

\end{array}
double f(double re, double im) {
        double r553527 = 0.5;
        double r553528 = 2.0;
        double r553529 = re;
        double r553530 = r553529 * r553529;
        double r553531 = im;
        double r553532 = r553531 * r553531;
        double r553533 = r553530 + r553532;
        double r553534 = sqrt(r553533);
        double r553535 = r553534 + r553529;
        double r553536 = r553528 * r553535;
        double r553537 = sqrt(r553536);
        double r553538 = r553527 * r553537;
        return r553538;
}
↓
double f(double re, double im) {
        double r553539 = re;
        double r553540 = -6.580886374598778e-301;
        bool r553541 = r553539 <= r553540;
        double r553542 = 2.0;
        double r553543 = im;
        double r553544 = 2.0;
        double r553545 = pow(r553543, r553544);
        double r553546 = pow(r553539, r553544);
        double r553547 = r553546 + r553545;
        double r553548 = sqrt(r553547);
        double r553549 = r553548 - r553539;
        double r553550 = r553545 / r553549;
        double r553551 = r553542 * r553550;
        double r553552 = sqrt(r553551);
        double r553553 = 0.5;
        double r553554 = r553552 * r553553;
        double r553555 = 6.674516804324248e+142;
        bool r553556 = r553539 <= r553555;
        double r553557 = r553539 * r553539;
        double r553558 = r553543 * r553543;
        double r553559 = r553557 + r553558;
        double r553560 = sqrt(r553559);
        double r553561 = r553560 + r553539;
        double r553562 = r553542 * r553561;
        double r553563 = sqrt(r553562);
        double r553564 = r553553 * r553563;
        double r553565 = r553544 * r553539;
        double r553566 = r553542 * r553565;
        double r553567 = sqrt(r553566);
        double r553568 = r553553 * r553567;
        double r553569 = r553556 ? r553564 : r553568;
        double r553570 = r553541 ? r553554 : r553569;
        return r553570;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.2
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -6.580886374598778e-301

    1. Initial program 46.4

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

    3. Applied add-exp-log48.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{e^{\log \left(re \cdot re + im \cdot im\right)}}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+48.5

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

    6. Simplified36.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{e^{\log \left(re \cdot re + im \cdot im\right)}} - re}}\]

    7. Simplified35.4

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

    if -6.580886374598778e-301 < re < 6.674516804324248e+142

    1. Initial program 20.5

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

    if 6.674516804324248e+142 < re

    1. Initial program 60.6

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{{re}^{2} + {im}^{2}} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))