Average Error: 39.4 → 11.7
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -16209155258582.345703125:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -16209155258582.345703125:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {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 r174738 = 0.5;
        double r174739 = 2.0;
        double r174740 = re;
        double r174741 = r174740 * r174740;
        double r174742 = im;
        double r174743 = r174742 * r174742;
        double r174744 = r174741 + r174743;
        double r174745 = sqrt(r174744);
        double r174746 = r174745 + r174740;
        double r174747 = r174739 * r174746;
        double r174748 = sqrt(r174747);
        double r174749 = r174738 * r174748;
        return r174749;
}
↓
double f(double re, double im) {
        double r174750 = re;
        double r174751 = -16209155258582.346;
        bool r174752 = r174750 <= r174751;
        double r174753 = 0.5;
        double r174754 = 2.0;
        double r174755 = 0.0;
        double r174756 = im;
        double r174757 = 2.0;
        double r174758 = pow(r174756, r174757);
        double r174759 = r174755 + r174758;
        double r174760 = hypot(r174750, r174756);
        double r174761 = r174760 - r174750;
        double r174762 = r174759 / r174761;
        double r174763 = r174754 * r174762;
        double r174764 = sqrt(r174763);
        double r174765 = r174753 * r174764;
        double r174766 = r174760 + r174750;
        double r174767 = r174754 * r174766;
        double r174768 = sqrt(r174767);
        double r174769 = r174753 * r174768;
        double r174770 = r174752 ? r174765 : r174769;
        return r174770;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.4
Target34.4
Herbie11.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -16209155258582.346

    1. Initial program 58.2

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

    3. Applied flip-+58.2

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

    4. Simplified42.4

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

    5. Simplified32.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]

    if -16209155258582.346 < re

    1. Initial program 33.3

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

    3. Applied hypot-def5.0

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -16209155258582.345703125:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))))