Average Error: 38.6 → 11.3
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.5530054912046227 \cdot 10^{-4}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.5530054912046227 \cdot 10^{-4}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\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 r20705 = 0.5;
        double r20706 = 2.0;
        double r20707 = re;
        double r20708 = r20707 * r20707;
        double r20709 = im;
        double r20710 = r20709 * r20709;
        double r20711 = r20708 + r20710;
        double r20712 = sqrt(r20711);
        double r20713 = r20712 - r20707;
        double r20714 = r20706 * r20713;
        double r20715 = sqrt(r20714);
        double r20716 = r20705 * r20715;
        return r20716;
}
↓
double f(double re, double im) {
        double r20717 = re;
        double r20718 = 0.00015530054912046227;
        bool r20719 = r20717 <= r20718;
        double r20720 = 0.5;
        double r20721 = 2.0;
        double r20722 = 1.0;
        double r20723 = im;
        double r20724 = hypot(r20717, r20723);
        double r20725 = r20724 - r20717;
        double r20726 = r20722 * r20725;
        double r20727 = r20721 * r20726;
        double r20728 = sqrt(r20727);
        double r20729 = r20720 * r20728;
        double r20730 = 2.0;
        double r20731 = pow(r20723, r20730);
        double r20732 = 0.0;
        double r20733 = r20731 + r20732;
        double r20734 = r20717 + r20724;
        double r20735 = r20733 / r20734;
        double r20736 = r20721 * r20735;
        double r20737 = sqrt(r20736);
        double r20738 = r20720 * r20737;
        double r20739 = r20719 ? r20729 : r20738;
        return r20739;
}

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 < 0.00015530054912046227

    1. Initial program 32.4

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

    3. Applied *-un-lft-identity32.4

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

    4. Applied *-un-lft-identity32.4

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

    5. Applied distribute-lft-out--32.4

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

    6. Simplified4.8

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

    if 0.00015530054912046227 < re

    1. Initial program 57.2

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

    3. Applied flip--57.2

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

    4. Simplified40.8

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

    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 simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.5530054912046227 \cdot 10^{-4}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))))