Average Error: 38.6 → 5.6
Time: 3.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.3168234763468002 \cdot 10^{-232}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.3168234763468002 \cdot 10^{-232}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\

\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 r183422 = 0.5;
        double r183423 = 2.0;
        double r183424 = re;
        double r183425 = r183424 * r183424;
        double r183426 = im;
        double r183427 = r183426 * r183426;
        double r183428 = r183425 + r183427;
        double r183429 = sqrt(r183428);
        double r183430 = r183429 + r183424;
        double r183431 = r183423 * r183430;
        double r183432 = sqrt(r183431);
        double r183433 = r183422 * r183432;
        return r183433;
}
↓
double f(double re, double im) {
        double r183434 = re;
        double r183435 = -4.3168234763468e-232;
        bool r183436 = r183434 <= r183435;
        double r183437 = 0.5;
        double r183438 = 2.0;
        double r183439 = im;
        double r183440 = 1.0;
        double r183441 = pow(r183439, r183440);
        double r183442 = hypot(r183434, r183439);
        double r183443 = r183442 - r183434;
        double r183444 = r183443 / r183439;
        double r183445 = r183441 / r183444;
        double r183446 = r183438 * r183445;
        double r183447 = sqrt(r183446);
        double r183448 = r183437 * r183447;
        double r183449 = r183440 * r183442;
        double r183450 = r183449 + r183434;
        double r183451 = r183438 * r183450;
        double r183452 = sqrt(r183451);
        double r183453 = r183437 * r183452;
        double r183454 = r183436 ? r183448 : r183453;
        return r183454;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -4.3168234763468e-232

    1. Initial program 47.9

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

    3. Applied flip-+47.8

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

    4. Simplified36.7

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

    5. Simplified31.3

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

    7. Applied sqr-pow31.3

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

    8. Applied associate-/l*12.2

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

    9. Simplified12.2

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

    if -4.3168234763468e-232 < re

    1. Initial program 31.5

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

    3. Applied *-un-lft-identity31.5

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

    4. Applied sqrt-prod31.5

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

    5. Simplified31.5

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

    6. Simplified0.5

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.3168234763468002 \cdot 10^{-232}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\mathsf{hypot}\left(re, im\right) - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))