Average Error: 39.2 → 6.4
Time: 5.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.0927697338173699 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.0927697338173699 \cdot 10^{-138}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r161732 = 0.5;
        double r161733 = 2.0;
        double r161734 = re;
        double r161735 = r161734 * r161734;
        double r161736 = im;
        double r161737 = r161736 * r161736;
        double r161738 = r161735 + r161737;
        double r161739 = sqrt(r161738);
        double r161740 = r161739 + r161734;
        double r161741 = r161733 * r161740;
        double r161742 = sqrt(r161741);
        double r161743 = r161732 * r161742;
        return r161743;
}
↓
double f(double re, double im) {
        double r161744 = re;
        double r161745 = -7.09276973381737e-138;
        bool r161746 = r161744 <= r161745;
        double r161747 = 0.5;
        double r161748 = 2.0;
        double r161749 = 1.0;
        double r161750 = im;
        double r161751 = hypot(r161744, r161750);
        double r161752 = r161751 - r161744;
        double r161753 = r161752 / r161750;
        double r161754 = r161750 / r161753;
        double r161755 = r161749 * r161754;
        double r161756 = r161748 * r161755;
        double r161757 = sqrt(r161756);
        double r161758 = r161747 * r161757;
        double r161759 = sqrt(r161749);
        double r161760 = r161759 * r161751;
        double r161761 = r161760 + r161744;
        double r161762 = r161748 * r161761;
        double r161763 = sqrt(r161762);
        double r161764 = r161747 * r161763;
        double r161765 = r161746 ? r161758 : r161764;
        return r161765;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.2
Target34.1
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -7.09276973381737e-138

    1. Initial program 52.6

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

    3. Applied flip-+52.6

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

    4. Simplified38.5

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

    5. Simplified31.1

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

    7. Applied *-un-lft-identity31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\color{blue}{1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}}\]

    8. Applied *-un-lft-identity31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{1 \cdot \left(0 + {im}^{2}\right)}}{1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\]

    9. Applied times-frac31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}\right)}}\]

    10. Simplified31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}\right)}\]

    11. Simplified14.3

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

    if -7.09276973381737e-138 < re

    1. Initial program 31.6

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

    3. Applied *-un-lft-identity31.6

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

    4. Applied sqrt-prod31.6

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

    5. Simplified1.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.0927697338173699 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))))