Average Error: 38.8 → 12.1
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\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 r213821 = 0.5;
        double r213822 = 2.0;
        double r213823 = re;
        double r213824 = r213823 * r213823;
        double r213825 = im;
        double r213826 = r213825 * r213825;
        double r213827 = r213824 + r213826;
        double r213828 = sqrt(r213827);
        double r213829 = r213828 + r213823;
        double r213830 = r213822 * r213829;
        double r213831 = sqrt(r213830);
        double r213832 = r213821 * r213831;
        return r213832;
}
↓
double f(double re, double im) {
        double r213833 = re;
        double r213834 = -9.189817498010996e+168;
        bool r213835 = r213833 <= r213834;
        double r213836 = 0.5;
        double r213837 = 2.0;
        double r213838 = im;
        double r213839 = r213838 * r213838;
        double r213840 = -1.0;
        double r213841 = hypot(r213833, r213838);
        double r213842 = fma(r213840, r213833, r213841);
        double r213843 = r213839 / r213842;
        double r213844 = r213837 * r213843;
        double r213845 = sqrt(r213844);
        double r213846 = r213836 * r213845;
        double r213847 = 1.0;
        double r213848 = sqrt(r213847);
        double r213849 = r213848 * r213841;
        double r213850 = r213849 + r213833;
        double r213851 = r213837 * r213850;
        double r213852 = sqrt(r213851);
        double r213853 = r213836 * r213852;
        double r213854 = r213835 ? r213846 : r213853;
        return r213854;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.8
Target33.8
Herbie12.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -9.189817498010996e+168

    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 flip-+64.0

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

    4. Simplified50.6

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

    5. Simplified31.9

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

    if -9.189817498010996e+168 < re

    1. Initial program 35.6

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

    3. Applied *-un-lft-identity35.6

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

    4. Applied sqrt-prod35.6

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

    5. Simplified9.6

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.189817498010996149444340978229122472326 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))))