Average Error: 38.7 → 12.8
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.4512261620833975 \cdot 10^{-30} \lor \neg \left(re \le 4.9019714476158837 \cdot 10^{154}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.4512261620833975 \cdot 10^{-30} \lor \neg \left(re \le 4.9019714476158837 \cdot 10^{154}\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\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 r16161 = 0.5;
        double r16162 = 2.0;
        double r16163 = re;
        double r16164 = r16163 * r16163;
        double r16165 = im;
        double r16166 = r16165 * r16165;
        double r16167 = r16164 + r16166;
        double r16168 = sqrt(r16167);
        double r16169 = r16168 - r16163;
        double r16170 = r16162 * r16169;
        double r16171 = sqrt(r16170);
        double r16172 = r16161 * r16171;
        return r16172;
}
↓
double f(double re, double im) {
        double r16173 = re;
        double r16174 = 1.4512261620833975e-30;
        bool r16175 = r16173 <= r16174;
        double r16176 = 4.9019714476158837e+154;
        bool r16177 = r16173 <= r16176;
        double r16178 = !r16177;
        bool r16179 = r16175 || r16178;
        double r16180 = 0.5;
        double r16181 = 2.0;
        double r16182 = 1.0;
        double r16183 = im;
        double r16184 = hypot(r16173, r16183);
        double r16185 = r16182 * r16184;
        double r16186 = r16185 - r16173;
        double r16187 = r16181 * r16186;
        double r16188 = sqrt(r16187);
        double r16189 = r16180 * r16188;
        double r16190 = 2.0;
        double r16191 = pow(r16183, r16190);
        double r16192 = 0.0;
        double r16193 = r16191 + r16192;
        double r16194 = r16173 + r16184;
        double r16195 = r16193 / r16194;
        double r16196 = r16181 * r16195;
        double r16197 = sqrt(r16196);
        double r16198 = r16180 * r16197;
        double r16199 = r16179 ? r16189 : r16198;
        return r16199;
}

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 < 1.4512261620833975e-30 or 4.9019714476158837e+154 < re

    1. Initial program 36.7

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

    3. Applied *-un-lft-identity36.7

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

    4. Applied sqrt-prod36.7

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

    5. Simplified36.7

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

    6. Simplified9.5

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

    if 1.4512261620833975e-30 < re < 4.9019714476158837e+154

    1. Initial program 49.9

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

    3. Applied flip--49.9

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

    4. Simplified31.0

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

    5. Simplified30.9

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.4512261620833975 \cdot 10^{-30} \lor \neg \left(re \le 4.9019714476158837 \cdot 10^{154}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +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)))))