Average Error: 38.7 → 11.3
Time: 3.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.40766140163207888 \cdot 10^{87} \lor \neg \left(re \le -9913.05910332904205 \lor \neg \left(re \le -4.026179673256086 \cdot 10^{-15}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.40766140163207888 \cdot 10^{87} \lor \neg \left(re \le -9913.05910332904205 \lor \neg \left(re \le -4.026179673256086 \cdot 10^{-15}\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 r202728 = 0.5;
        double r202729 = 2.0;
        double r202730 = re;
        double r202731 = r202730 * r202730;
        double r202732 = im;
        double r202733 = r202732 * r202732;
        double r202734 = r202731 + r202733;
        double r202735 = sqrt(r202734);
        double r202736 = r202735 + r202730;
        double r202737 = r202729 * r202736;
        double r202738 = sqrt(r202737);
        double r202739 = r202728 * r202738;
        return r202739;
}
↓
double f(double re, double im) {
        double r202740 = re;
        double r202741 = -4.407661401632079e+87;
        bool r202742 = r202740 <= r202741;
        double r202743 = -9913.059103329042;
        bool r202744 = r202740 <= r202743;
        double r202745 = -4.026179673256086e-15;
        bool r202746 = r202740 <= r202745;
        double r202747 = !r202746;
        bool r202748 = r202744 || r202747;
        double r202749 = !r202748;
        bool r202750 = r202742 || r202749;
        double r202751 = 0.5;
        double r202752 = 2.0;
        double r202753 = im;
        double r202754 = 2.0;
        double r202755 = pow(r202753, r202754);
        double r202756 = hypot(r202740, r202753);
        double r202757 = r202756 - r202740;
        double r202758 = r202755 / r202757;
        double r202759 = r202752 * r202758;
        double r202760 = sqrt(r202759);
        double r202761 = r202751 * r202760;
        double r202762 = r202756 + r202740;
        double r202763 = r202752 * r202762;
        double r202764 = sqrt(r202763);
        double r202765 = r202751 * r202764;
        double r202766 = r202750 ? r202761 : r202765;
        return r202766;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.7
Target33.7
Herbie11.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -4.407661401632079e+87 or -9913.059103329042 < re < -4.026179673256086e-15

    1. Initial program 59.4

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

    3. Applied flip-+59.4

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

    4. Simplified43.8

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

    5. Simplified31.0

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

    if -4.407661401632079e+87 < re < -9913.059103329042 or -4.026179673256086e-15 < re

    1. Initial program 33.8

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

    3. Applied hypot-def6.7

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.40766140163207888 \cdot 10^{87} \lor \neg \left(re \le -9913.05910332904205 \lor \neg \left(re \le -4.026179673256086 \cdot 10^{-15}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))))