Average Error: 38.6 → 30.2
Time: 19.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.847894715956925849700723533852013544459 \cdot 10^{109}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.847894715956925849700723533852013544459 \cdot 10^{109}:\\
\;\;\;\;0\\

\mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\
\;\;\;\;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 r144445 = 0.5;
        double r144446 = 2.0;
        double r144447 = re;
        double r144448 = r144447 * r144447;
        double r144449 = im;
        double r144450 = r144449 * r144449;
        double r144451 = r144448 + r144450;
        double r144452 = sqrt(r144451);
        double r144453 = r144452 + r144447;
        double r144454 = r144446 * r144453;
        double r144455 = sqrt(r144454);
        double r144456 = r144445 * r144455;
        return r144456;
}
↓
double f(double re, double im) {
        double r144457 = re;
        double r144458 = -6.847894715956926e+109;
        bool r144459 = r144457 <= r144458;
        double r144460 = 0.0;
        double r144461 = 1.3595155319523303e+138;
        bool r144462 = r144457 <= r144461;
        double r144463 = 0.5;
        double r144464 = 2.0;
        double r144465 = r144457 * r144457;
        double r144466 = im;
        double r144467 = r144466 * r144466;
        double r144468 = r144465 + r144467;
        double r144469 = sqrt(r144468);
        double r144470 = r144469 + r144457;
        double r144471 = r144464 * r144470;
        double r144472 = sqrt(r144471);
        double r144473 = r144463 * r144472;
        double r144474 = 2.0;
        double r144475 = r144474 * r144457;
        double r144476 = r144464 * r144475;
        double r144477 = sqrt(r144476);
        double r144478 = r144463 * r144477;
        double r144479 = r144462 ? r144473 : r144478;
        double r144480 = r144459 ? r144460 : r144479;
        return r144480;
}

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
Target33.7
Herbie30.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -6.847894715956926e+109

    1. Initial program 61.1

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

    3. Applied add-sqr-sqrt61.1

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

    4. Applied sqrt-prod62.1

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

    5. Taylor expanded around -inf 51.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}}\]

    if -6.847894715956926e+109 < re < 1.3595155319523303e+138

    1. Initial program 29.4

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

    if 1.3595155319523303e+138 < re

    1. Initial program 58.8

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification30.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.847894715956925849700723533852013544459 \cdot 10^{109}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))