Average Error: 39.1 → 14.2
Time: 7.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\
\;\;\;\;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 r132 = 0.5;
        double r133 = 2.0;
        double r134 = re;
        double r135 = r134 * r134;
        double r136 = im;
        double r137 = r136 * r136;
        double r138 = r135 + r137;
        double r139 = sqrt(r138);
        double r140 = r139 - r134;
        double r141 = r133 * r140;
        double r142 = sqrt(r141);
        double r143 = r132 * r142;
        return r143;
}
↓
double f(double re, double im) {
        double r144 = im;
        double r145 = -4.1789077984601566e+114;
        bool r146 = r144 <= r145;
        double r147 = -8.34056697677881e+54;
        bool r148 = r144 <= r147;
        double r149 = 8.575500542659089e-157;
        bool r150 = r144 <= r149;
        double r151 = 3.3371304803298494e+120;
        bool r152 = r144 <= r151;
        double r153 = !r152;
        bool r154 = r150 || r153;
        double r155 = !r154;
        bool r156 = r148 || r155;
        double r157 = !r156;
        bool r158 = r146 || r157;
        double r159 = 0.5;
        double r160 = 2.0;
        double r161 = re;
        double r162 = hypot(r161, r144);
        double r163 = r162 - r161;
        double r164 = 0.0;
        double r165 = r163 + r164;
        double r166 = r160 * r165;
        double r167 = sqrt(r166);
        double r168 = r159 * r167;
        double r169 = 2.0;
        double r170 = pow(r144, r169);
        double r171 = r170 + r164;
        double r172 = r161 + r162;
        double r173 = r171 / r172;
        double r174 = r160 * r173;
        double r175 = sqrt(r174);
        double r176 = r159 * r175;
        double r177 = r158 ? r168 : r176;
        return r177;
}

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 im < -4.1789077984601566e+114 or -8.34056697677881e+54 < im < 8.575500542659089e-157 or 3.3371304803298494e+120 < im

    1. Initial program 44.5

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

    3. Applied add-cube-cbrt45.2

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

    4. Applied add-sqr-sqrt45.2

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

    5. Applied sqrt-prod45.2

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

    6. Applied prod-diff45.3

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

    7. Simplified15.4

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

    8. Simplified12.5

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

    if -4.1789077984601566e+114 < im < -8.34056697677881e+54 or 8.575500542659089e-157 < im < 3.3371304803298494e+120

    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--31.6

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

    4. Simplified24.1

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

    5. Simplified19.3

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

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