Average Error: 38.6 → 11.6
Time: 4.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -0.4962378853117394:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -0.4962378853117394:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

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

\end{array}
double f(double re, double im) {
        double r231561 = 0.5;
        double r231562 = 2.0;
        double r231563 = re;
        double r231564 = r231563 * r231563;
        double r231565 = im;
        double r231566 = r231565 * r231565;
        double r231567 = r231564 + r231566;
        double r231568 = sqrt(r231567);
        double r231569 = r231568 + r231563;
        double r231570 = r231562 * r231569;
        double r231571 = sqrt(r231570);
        double r231572 = r231561 * r231571;
        return r231572;
}
↓
double f(double re, double im) {
        double r231573 = re;
        double r231574 = -0.4962378853117394;
        bool r231575 = r231573 <= r231574;
        double r231576 = 0.5;
        double r231577 = 2.0;
        double r231578 = 0.0;
        double r231579 = im;
        double r231580 = 2.0;
        double r231581 = pow(r231579, r231580);
        double r231582 = r231578 + r231581;
        double r231583 = hypot(r231573, r231579);
        double r231584 = r231583 - r231573;
        double r231585 = r231582 / r231584;
        double r231586 = r231577 * r231585;
        double r231587 = sqrt(r231586);
        double r231588 = r231576 * r231587;
        double r231589 = sqrt(r231577);
        double r231590 = 1.0;
        double r231591 = sqrt(r231590);
        double r231592 = r231591 * r231583;
        double r231593 = r231592 + r231573;
        double r231594 = sqrt(r231593);
        double r231595 = r231589 * r231594;
        double r231596 = r231576 * r231595;
        double r231597 = r231575 ? r231588 : r231596;
        return r231597;
}

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
Target33.1
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -0.4962378853117394

    1. Initial program 57.4

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

    3. Applied flip-+57.4

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

    4. Simplified39.6

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

    5. Simplified30.3

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

    if -0.4962378853117394 < re

    1. Initial program 32.2

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

    3. Applied *-un-lft-identity32.2

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

    4. Applied sqrt-prod32.2

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

    5. Simplified5.0

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

    7. Applied sqrt-prod5.3

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.4962378853117394:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))))