Average Error: 38.9 → 11.9
Time: 3.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\
\;\;\;\;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(\mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r118604 = 0.5;
        double r118605 = 2.0;
        double r118606 = re;
        double r118607 = r118606 * r118606;
        double r118608 = im;
        double r118609 = r118608 * r118608;
        double r118610 = r118607 + r118609;
        double r118611 = sqrt(r118610);
        double r118612 = r118611 + r118606;
        double r118613 = r118605 * r118612;
        double r118614 = sqrt(r118613);
        double r118615 = r118604 * r118614;
        return r118615;
}
↓
double f(double re, double im) {
        double r118616 = re;
        double r118617 = -5.904936880911102e+122;
        bool r118618 = r118616 <= r118617;
        double r118619 = 0.5;
        double r118620 = 2.0;
        double r118621 = im;
        double r118622 = r118621 * r118621;
        double r118623 = -1.0;
        double r118624 = hypot(r118616, r118621);
        double r118625 = fma(r118623, r118616, r118624);
        double r118626 = r118622 / r118625;
        double r118627 = r118620 * r118626;
        double r118628 = sqrt(r118627);
        double r118629 = r118619 * r118628;
        double r118630 = r118624 + r118616;
        double r118631 = r118620 * r118630;
        double r118632 = sqrt(r118631);
        double r118633 = r118619 * r118632;
        double r118634 = r118618 ? r118629 : r118633;
        return r118634;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.9
Target33.8
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -5.904936880911102e+122

    1. Initial program 62.2

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

    3. Applied flip-+62.2

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

    4. Simplified47.7

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

    5. Simplified31.7

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

    if -5.904936880911102e+122 < re

    1. Initial program 34.6

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

    3. Applied hypot-def8.3

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))))