Average Error: 38.7 → 5.7
Time: 3.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.1917524124082504 \cdot 10^{-219}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.1917524124082504 \cdot 10^{-219}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\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 r218007 = 0.5;
        double r218008 = 2.0;
        double r218009 = re;
        double r218010 = r218009 * r218009;
        double r218011 = im;
        double r218012 = r218011 * r218011;
        double r218013 = r218010 + r218012;
        double r218014 = sqrt(r218013);
        double r218015 = r218014 + r218009;
        double r218016 = r218008 * r218015;
        double r218017 = sqrt(r218016);
        double r218018 = r218007 * r218017;
        return r218018;
}
↓
double f(double re, double im) {
        double r218019 = re;
        double r218020 = -9.19175241240825e-219;
        bool r218021 = r218019 <= r218020;
        double r218022 = 0.5;
        double r218023 = 2.0;
        double r218024 = im;
        double r218025 = -1.0;
        double r218026 = hypot(r218019, r218024);
        double r218027 = fma(r218025, r218019, r218026);
        double r218028 = r218024 / r218027;
        double r218029 = r218024 * r218028;
        double r218030 = r218023 * r218029;
        double r218031 = sqrt(r218030);
        double r218032 = r218022 * r218031;
        double r218033 = 1.0;
        double r218034 = sqrt(r218033);
        double r218035 = r218034 * r218026;
        double r218036 = r218035 + r218019;
        double r218037 = r218023 * r218036;
        double r218038 = sqrt(r218037);
        double r218039 = r218022 * r218038;
        double r218040 = r218021 ? r218032 : r218039;
        return r218040;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.7
Target33.7
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -9.19175241240825e-219

    1. Initial program 48.2

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

    3. Applied flip-+48.1

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

    4. Simplified36.8

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

    5. Simplified31.2

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

    7. Applied *-un-lft-identity31.2

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

    8. Applied times-frac12.5

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

    9. Simplified12.5

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

    if -9.19175241240825e-219 < re

    1. Initial program 31.3

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

    3. Applied *-un-lft-identity31.3

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

    4. Applied sqrt-prod31.3

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

    5. Simplified0.5

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.1917524124082504 \cdot 10^{-219}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))))