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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.9400061054300293 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}

↓

\begin{array}{l}
\mathbf{if}\;re \le -4.9400061054300293 \cdot 10^{-58}:\\
\;\;\;\;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 r240456 = 0.5;
        double r240457 = 2.0;
        double r240458 = re;
        double r240459 = r240458 * r240458;
        double r240460 = im;
        double r240461 = r240460 * r240460;
        double r240462 = r240459 + r240461;
        double r240463 = sqrt(r240462);
        double r240464 = r240463 + r240458;
        double r240465 = r240457 * r240464;
        double r240466 = sqrt(r240465);
        double r240467 = r240456 * r240466;
        return r240467;
}

↓

double f(double re, double im) {
        double r240468 = re;
        double r240469 = -4.940006105430029e-58;
        bool r240470 = r240468 <= r240469;
        double r240471 = 0.5;
        double r240472 = 2.0;
        double r240473 = 0.0;
        double r240474 = im;
        double r240475 = 2.0;
        double r240476 = pow(r240474, r240475);
        double r240477 = r240473 + r240476;
        double r240478 = hypot(r240468, r240474);
        double r240479 = r240478 - r240468;
        double r240480 = r240477 / r240479;
        double r240481 = r240472 * r240480;
        double r240482 = sqrt(r240481);
        double r240483 = r240471 * r240482;
        double r240484 = r240478 + r240468;
        double r240485 = r240472 * r240484;
        double r240486 = sqrt(r240485);
        double r240487 = r240471 * r240486;
        double r240488 = r240470 ? r240483 : r240487;
        return r240488;
}

Original	38.8
Target	33.7
Herbie	11.3

Derivation

Split input into 2 regimes
if re < -4.940006105430029e-58
1. Initial program 54.9
  \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
2. Using strategy rm
3. Applied flip-+54.9
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
4. Simplified39.1
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
5. Simplified29.9
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]
if -4.940006105430029e-58 < re
1. Initial program 31.9
  \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
2. Using strategy rm
3. Applied hypot-def3.4
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]
Recombined 2 regimes into one program.
Final simplification11.3
\[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.9400061054300293 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))))

Error

Try it out

Target

Derivation

`if re < -4.940006105430029e-58`

`if -4.940006105430029e-58 < re`

Reproduce

In
Out