Average Error: 38.6 → 13.5
Time: 3.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 1.88481914151467382 \cdot 10^{-199}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 1.03470872006332819 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{elif}\;im \cdot im \le 1.0159716000531194 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{elif}\;im \cdot im \le 3.04445915423810877 \cdot 10^{195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 9.405231814021979 \cdot 10^{301}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\\ \end{array}\]
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 1.88481914151467382 \cdot 10^{-199}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{elif}\;im \cdot im \le 1.03470872006332819 \cdot 10^{-159}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

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

\mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\mathbf{elif}\;im \cdot im \le 3.04445915423810877 \cdot 10^{195}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{elif}\;im \cdot im \le 9.405231814021979 \cdot 10^{301}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\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 r18240 = 0.5;
        double r18241 = 2.0;
        double r18242 = re;
        double r18243 = r18242 * r18242;
        double r18244 = im;
        double r18245 = r18244 * r18244;
        double r18246 = r18243 + r18245;
        double r18247 = sqrt(r18246);
        double r18248 = r18247 - r18242;
        double r18249 = r18241 * r18248;
        double r18250 = sqrt(r18249);
        double r18251 = r18240 * r18250;
        return r18251;
}
↓
double f(double re, double im) {
        double r18252 = im;
        double r18253 = r18252 * r18252;
        double r18254 = 1.8848191415146738e-199;
        bool r18255 = r18253 <= r18254;
        double r18256 = 0.5;
        double r18257 = 2.0;
        double r18258 = re;
        double r18259 = hypot(r18258, r18252);
        double r18260 = r18259 - r18258;
        double r18261 = 0.0;
        double r18262 = r18260 + r18261;
        double r18263 = r18257 * r18262;
        double r18264 = sqrt(r18263);
        double r18265 = r18256 * r18264;
        double r18266 = 1.0347087200633282e-159;
        bool r18267 = r18253 <= r18266;
        double r18268 = 2.0;
        double r18269 = pow(r18252, r18268);
        double r18270 = r18269 + r18261;
        double r18271 = r18258 + r18259;
        double r18272 = r18270 / r18271;
        double r18273 = r18257 * r18272;
        double r18274 = sqrt(r18273);
        double r18275 = r18256 * r18274;
        double r18276 = 1.0159716000531194e-52;
        bool r18277 = r18253 <= r18276;
        double r18278 = 8.59737452451693e+71;
        bool r18279 = r18253 <= r18278;
        double r18280 = 3.044459154238109e+195;
        bool r18281 = r18253 <= r18280;
        double r18282 = 9.40523181402198e+301;
        bool r18283 = r18253 <= r18282;
        double r18284 = sqrt(r18257);
        double r18285 = sqrt(r18260);
        double r18286 = r18284 * r18285;
        double r18287 = r18256 * r18286;
        double r18288 = r18283 ? r18275 : r18287;
        double r18289 = r18281 ? r18265 : r18288;
        double r18290 = r18279 ? r18275 : r18289;
        double r18291 = r18277 ? r18265 : r18290;
        double r18292 = r18267 ? r18275 : r18291;
        double r18293 = r18255 ? r18265 : r18292;
        return r18293;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* im im) < 1.8848191415146738e-199 or 1.0347087200633282e-159 < (* im im) < 1.0159716000531194e-52 or 8.59737452451693e+71 < (* im im) < 3.044459154238109e+195

    1. Initial program 33.4

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

    3. Applied add-cube-cbrt34.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - \color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}\right)}\]

    4. Applied add-sqr-sqrt34.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    5. Applied sqrt-prod34.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    6. Applied prod-diff34.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}, \sqrt{\sqrt{re \cdot re + im \cdot im}}, -\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}}\]

    7. Simplified21.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)} + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}\]

    8. Simplified17.8

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

    if 1.8848191415146738e-199 < (* im im) < 1.0347087200633282e-159 or 1.0159716000531194e-52 < (* im im) < 8.59737452451693e+71 or 3.044459154238109e+195 < (* im im) < 9.40523181402198e+301

    1. Initial program 22.0

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

    3. Applied flip--27.8

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

    4. Simplified21.2

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

    5. Simplified14.6

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

    if 9.40523181402198e+301 < (* im im)

    1. Initial program 63.2

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

    3. Applied add-cube-cbrt63.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - \color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}\right)}\]

    4. Applied add-sqr-sqrt63.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    5. Applied sqrt-prod63.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    6. Applied prod-diff63.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}, \sqrt{\sqrt{re \cdot re + im \cdot im}}, -\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}}\]

    7. Simplified3.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)} + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}\]

    8. Simplified3.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + \color{blue}{0}\right)}\]
    9. Using strategy rm

    10. Applied sqrt-prod4.2

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

    11. Simplified4.2

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 1.88481914151467382 \cdot 10^{-199}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 1.03470872006332819 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{elif}\;im \cdot im \le 1.0159716000531194 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 8.5973745245169301 \cdot 10^{71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{elif}\;im \cdot im \le 3.04445915423810877 \cdot 10^{195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{elif}\;im \cdot im \le 9.405231814021979 \cdot 10^{301}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))