Average Error: 38.4 → 14.0
Time: 3.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 6.69696849832756618 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.88481914151467382 \cdot 10^{-199}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;im \cdot im \le 3.330023461604586 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.0159716000531194 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 4.4483345476132472 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \le 6.69696849832756618 \cdot 10^{-270}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\

\mathbf{elif}\;im \cdot im \le 1.88481914151467382 \cdot 10^{-199}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{elif}\;im \cdot im \le 3.330023461604586 \cdot 10^{-91}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\

\mathbf{elif}\;im \cdot im \le 1.0159716000531194 \cdot 10^{-52}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\

\mathbf{elif}\;im \cdot im \le 4.4483345476132472 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

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

\end{array}
double f(double re, double im) {
        double r154142 = 0.5;
        double r154143 = 2.0;
        double r154144 = re;
        double r154145 = r154144 * r154144;
        double r154146 = im;
        double r154147 = r154146 * r154146;
        double r154148 = r154145 + r154147;
        double r154149 = sqrt(r154148);
        double r154150 = r154149 + r154144;
        double r154151 = r154143 * r154150;
        double r154152 = sqrt(r154151);
        double r154153 = r154142 * r154152;
        return r154153;
}
↓
double f(double re, double im) {
        double r154154 = im;
        double r154155 = r154154 * r154154;
        double r154156 = 6.696968498327566e-270;
        bool r154157 = r154155 <= r154156;
        double r154158 = 0.5;
        double r154159 = 2.0;
        double r154160 = re;
        double r154161 = hypot(r154160, r154154);
        double r154162 = r154161 + r154160;
        double r154163 = r154159 * r154162;
        double r154164 = sqrt(r154163);
        double r154165 = r154158 * r154164;
        double r154166 = 1.8848191415146738e-199;
        bool r154167 = r154155 <= r154166;
        double r154168 = 0.0;
        double r154169 = 2.0;
        double r154170 = pow(r154154, r154169);
        double r154171 = r154168 + r154170;
        double r154172 = r154161 - r154160;
        double r154173 = r154171 / r154172;
        double r154174 = r154159 * r154173;
        double r154175 = sqrt(r154174);
        double r154176 = r154158 * r154175;
        double r154177 = 3.330023461604586e-91;
        bool r154178 = r154155 <= r154177;
        double r154179 = 1.0159716000531194e-52;
        bool r154180 = r154155 <= r154179;
        double r154181 = 8.59737452451693e+71;
        bool r154182 = r154155 <= r154181;
        double r154183 = 4.448334547613247e+154;
        bool r154184 = r154155 <= r154183;
        double r154185 = sqrt(r154159);
        double r154186 = sqrt(r154162);
        double r154187 = r154185 * r154186;
        double r154188 = r154158 * r154187;
        double r154189 = r154184 ? r154176 : r154188;
        double r154190 = r154182 ? r154165 : r154189;
        double r154191 = r154180 ? r154176 : r154190;
        double r154192 = r154178 ? r154165 : r154191;
        double r154193 = r154167 ? r154176 : r154192;
        double r154194 = r154157 ? r154165 : r154193;
        return r154194;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.4
Herbie14.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* im im) < 6.696968498327566e-270 or 1.8848191415146738e-199 < (* im im) < 3.330023461604586e-91 or 1.0159716000531194e-52 < (* im im) < 8.59737452451693e+71

    1. Initial program 34.7

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

    3. Applied hypot-def17.6

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

    if 6.696968498327566e-270 < (* im im) < 1.8848191415146738e-199 or 3.330023461604586e-91 < (* im im) < 1.0159716000531194e-52 or 8.59737452451693e+71 < (* im im) < 4.448334547613247e+154

    1. Initial program 23.4

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

    3. Applied flip-+32.4

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

    4. Simplified24.4

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

    5. Simplified20.6

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

    if 4.448334547613247e+154 < (* im im)

    1. Initial program 49.1

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

    3. Applied hypot-def6.4

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

    5. Applied sqrt-prod6.8

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

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 6.69696849832756618 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.88481914151467382 \cdot 10^{-199}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;im \cdot im \le 3.330023461604586 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.0159716000531194 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{elif}\;im \cdot im \le 4.4483345476132472 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))))