Average Error: 38.0 → 11.2
Time: 3.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\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 r13053 = 0.5;
        double r13054 = 2.0;
        double r13055 = re;
        double r13056 = r13055 * r13055;
        double r13057 = im;
        double r13058 = r13057 * r13057;
        double r13059 = r13056 + r13058;
        double r13060 = sqrt(r13059);
        double r13061 = r13060 - r13055;
        double r13062 = r13054 * r13061;
        double r13063 = sqrt(r13062);
        double r13064 = r13053 * r13063;
        return r13064;
}
↓
double f(double re, double im) {
        double r13065 = re;
        double r13066 = 3.226330986754208e-16;
        bool r13067 = r13065 <= r13066;
        double r13068 = 0.5;
        double r13069 = 2.0;
        double r13070 = im;
        double r13071 = hypot(r13065, r13070);
        double r13072 = r13071 - r13065;
        double r13073 = r13069 * r13072;
        double r13074 = sqrt(r13073);
        double r13075 = r13068 * r13074;
        double r13076 = 2.0;
        double r13077 = pow(r13070, r13076);
        double r13078 = 0.0;
        double r13079 = r13077 + r13078;
        double r13080 = r13065 + r13071;
        double r13081 = r13079 / r13080;
        double r13082 = r13069 * r13081;
        double r13083 = sqrt(r13082);
        double r13084 = r13068 * r13083;
        double r13085 = r13067 ? r13075 : r13084;
        return r13085;
}

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 < 3.226330986754208e-16

    1. Initial program 31.4

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

    3. Applied hypot-def4.4

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

    if 3.226330986754208e-16 < re

    1. Initial program 56.4

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

    3. Applied flip--56.4

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

    4. Simplified39.5

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

    5. Simplified30.1

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))))