Average Error: 38.4 → 11.8
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.31430247975949898 \cdot 10^{228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;re \le -3.2591409009285863 \cdot 10^{213}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right) + re\right)}\\ \mathbf{elif}\;re \le -1.31134963704057172 \cdot 10^{31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.31430247975949898 \cdot 10^{228}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{elif}\;re \le -3.2591409009285863 \cdot 10^{213}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right) + re\right)}\\

\mathbf{elif}\;re \le -1.31134963704057172 \cdot 10^{31}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

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

\end{array}
double f(double re, double im) {
        double r286819 = 0.5;
        double r286820 = 2.0;
        double r286821 = re;
        double r286822 = r286821 * r286821;
        double r286823 = im;
        double r286824 = r286823 * r286823;
        double r286825 = r286822 + r286824;
        double r286826 = sqrt(r286825);
        double r286827 = r286826 + r286821;
        double r286828 = r286820 * r286827;
        double r286829 = sqrt(r286828);
        double r286830 = r286819 * r286829;
        return r286830;
}
↓
double f(double re, double im) {
        double r286831 = re;
        double r286832 = -7.314302479759499e+228;
        bool r286833 = r286831 <= r286832;
        double r286834 = 0.5;
        double r286835 = 2.0;
        double r286836 = im;
        double r286837 = r286836 * r286836;
        double r286838 = hypot(r286831, r286836);
        double r286839 = r286838 - r286831;
        double r286840 = r286837 / r286839;
        double r286841 = r286835 * r286840;
        double r286842 = sqrt(r286841);
        double r286843 = r286834 * r286842;
        double r286844 = -3.2591409009285863e+213;
        bool r286845 = r286831 <= r286844;
        double r286846 = 1.0;
        double r286847 = sqrt(r286838);
        double r286848 = r286847 * r286847;
        double r286849 = r286846 * r286848;
        double r286850 = r286849 + r286831;
        double r286851 = r286835 * r286850;
        double r286852 = sqrt(r286851);
        double r286853 = r286834 * r286852;
        double r286854 = -1.3113496370405717e+31;
        bool r286855 = r286831 <= r286854;
        double r286856 = r286846 * r286838;
        double r286857 = r286856 + r286831;
        double r286858 = r286835 * r286857;
        double r286859 = sqrt(r286858);
        double r286860 = r286834 * r286859;
        double r286861 = r286855 ? r286843 : r286860;
        double r286862 = r286845 ? r286853 : r286861;
        double r286863 = r286833 ? r286843 : r286862;
        return r286863;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.1
Herbie11.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -7.314302479759499e+228 or -3.2591409009285863e+213 < re < -1.3113496370405717e+31

    1. Initial program 57.3

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

    3. Applied flip-+57.3

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

    4. Simplified40.6

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

    5. Simplified30.5

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

    if -7.314302479759499e+228 < re < -3.2591409009285863e+213

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied sqrt-prod64.0

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

    5. Simplified64.0

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

    6. Simplified41.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt48.2

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

    if -1.3113496370405717e+31 < re

    1. Initial program 32.7

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

    3. Applied *-un-lft-identity32.7

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

    4. Applied sqrt-prod32.7

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

    5. Simplified32.7

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

    6. Simplified6.0

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.31430247975949898 \cdot 10^{228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;re \le -3.2591409009285863 \cdot 10^{213}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right) + re\right)}\\ \mathbf{elif}\;re \le -1.31134963704057172 \cdot 10^{31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))))