Average Error: 38.2 → 12.5
Time: 5.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.9670001249123335 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.9670001249123335 \cdot 10^{-147}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r20233 = 0.5;
        double r20234 = 2.0;
        double r20235 = re;
        double r20236 = r20235 * r20235;
        double r20237 = im;
        double r20238 = r20237 * r20237;
        double r20239 = r20236 + r20238;
        double r20240 = sqrt(r20239);
        double r20241 = r20240 - r20235;
        double r20242 = r20234 * r20241;
        double r20243 = sqrt(r20242);
        double r20244 = r20233 * r20243;
        return r20244;
}
↓
double f(double re, double im) {
        double r20245 = re;
        double r20246 = 1.9670001249123335e-147;
        bool r20247 = r20245 <= r20246;
        double r20248 = 0.5;
        double r20249 = 2.0;
        double r20250 = im;
        double r20251 = hypot(r20245, r20250);
        double r20252 = r20251 - r20245;
        double r20253 = 0.0;
        double r20254 = r20252 + r20253;
        double r20255 = r20249 * r20254;
        double r20256 = sqrt(r20255);
        double r20257 = r20248 * r20256;
        double r20258 = 2.0;
        double r20259 = pow(r20250, r20258);
        double r20260 = r20259 + r20253;
        double r20261 = r20245 + r20251;
        double r20262 = r20260 / r20261;
        double r20263 = r20249 * r20262;
        double r20264 = sqrt(r20263);
        double r20265 = r20248 * r20264;
        double r20266 = r20247 ? r20257 : r20265;
        return r20266;
}

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 2 regimes

  2. if re < 1.9670001249123335e-147

    1. Initial program 30.7

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

    3. Applied add-cube-cbrt30.8

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

    4. Applied add-sqr-sqrt30.8

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

    5. Applied sqrt-prod30.9

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

    6. Applied prod-diff30.9

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

    7. Simplified1.7

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

    8. Simplified1.7

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

    if 1.9670001249123335e-147 < re

    1. Initial program 51.1

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

    3. Applied flip--51.1

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

    4. Simplified37.2

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

    5. Simplified31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.9670001249123335 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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)))))