Average Error: 38.3 → 12.0
Time: 3.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

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

\end{array}
double f(double re, double im) {
        double r256716 = 0.5;
        double r256717 = 2.0;
        double r256718 = re;
        double r256719 = r256718 * r256718;
        double r256720 = im;
        double r256721 = r256720 * r256720;
        double r256722 = r256719 + r256721;
        double r256723 = sqrt(r256722);
        double r256724 = r256723 + r256718;
        double r256725 = r256717 * r256724;
        double r256726 = sqrt(r256725);
        double r256727 = r256716 * r256726;
        return r256727;
}
↓
double f(double re, double im) {
        double r256728 = re;
        double r256729 = -1.7761354360071877e+40;
        bool r256730 = r256728 <= r256729;
        double r256731 = -1.0136200979503546e-07;
        bool r256732 = r256728 <= r256731;
        double r256733 = -1.941232155667052e-71;
        bool r256734 = r256728 <= r256733;
        double r256735 = !r256734;
        bool r256736 = r256732 || r256735;
        double r256737 = !r256736;
        bool r256738 = r256730 || r256737;
        double r256739 = 0.5;
        double r256740 = 2.0;
        double r256741 = im;
        double r256742 = 2.0;
        double r256743 = pow(r256741, r256742);
        double r256744 = hypot(r256728, r256741);
        double r256745 = r256744 - r256728;
        double r256746 = r256743 / r256745;
        double r256747 = r256740 * r256746;
        double r256748 = sqrt(r256747);
        double r256749 = r256739 * r256748;
        double r256750 = r256744 + r256728;
        double r256751 = r256740 * r256750;
        double r256752 = sqrt(r256751);
        double r256753 = r256739 * r256752;
        double r256754 = r256738 ? r256749 : r256753;
        return r256754;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.4
Herbie12.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -1.7761354360071877e+40 or -1.0136200979503546e-07 < re < -1.941232155667052e-71

    1. Initial program 54.9

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

    3. Applied flip-+54.9

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

    4. Simplified40.0

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

    5. Simplified31.7

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

    if -1.7761354360071877e+40 < re < -1.0136200979503546e-07 or -1.941232155667052e-71 < re

    1. Initial program 32.1

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

    3. Applied hypot-def4.6

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))