Average Error: 38.8 → 12.1
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r215413 = 0.5;
        double r215414 = 2.0;
        double r215415 = re;
        double r215416 = r215415 * r215415;
        double r215417 = im;
        double r215418 = r215417 * r215417;
        double r215419 = r215416 + r215418;
        double r215420 = sqrt(r215419);
        double r215421 = r215420 + r215415;
        double r215422 = r215414 * r215421;
        double r215423 = sqrt(r215422);
        double r215424 = r215413 * r215423;
        return r215424;
}
↓
double f(double re, double im) {
        double r215425 = re;
        double r215426 = -9.189817498010996e+168;
        bool r215427 = r215425 <= r215426;
        double r215428 = 0.5;
        double r215429 = 2.0;
        double r215430 = im;
        double r215431 = r215430 * r215430;
        double r215432 = -1.0;
        double r215433 = hypot(r215425, r215430);
        double r215434 = fma(r215432, r215425, r215433);
        double r215435 = r215431 / r215434;
        double r215436 = r215429 * r215435;
        double r215437 = sqrt(r215436);
        double r215438 = r215428 * r215437;
        double r215439 = 1.0;
        double r215440 = sqrt(r215439);
        double r215441 = r215440 * r215433;
        double r215442 = r215441 + r215425;
        double r215443 = r215429 * r215442;
        double r215444 = sqrt(r215443);
        double r215445 = r215428 * r215444;
        double r215446 = r215427 ? r215438 : r215445;
        return r215446;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.8
Target33.8
Herbie12.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -9.189817498010996e+168

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Simplified50.6

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

    5. Simplified31.9

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

    if -9.189817498010996e+168 < re

    1. Initial program 35.6

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

    3. Applied *-un-lft-identity35.6

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

    4. Applied sqrt-prod35.6

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

    5. Simplified9.6

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))))