Average Error: 38.9 → 11.9
Time: 7.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\
\;\;\;\;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 r754 = 0.5;
        double r755 = 2.0;
        double r756 = re;
        double r757 = r756 * r756;
        double r758 = im;
        double r759 = r758 * r758;
        double r760 = r757 + r759;
        double r761 = sqrt(r760);
        double r762 = r761 + r756;
        double r763 = r755 * r762;
        double r764 = sqrt(r763);
        double r765 = r754 * r764;
        return r765;
}
↓
double f(double re, double im) {
        double r766 = re;
        double r767 = -5.904936880911102e+122;
        bool r768 = r766 <= r767;
        double r769 = 0.5;
        double r770 = 2.0;
        double r771 = im;
        double r772 = r771 * r771;
        double r773 = -1.0;
        double r774 = hypot(r766, r771);
        double r775 = fma(r773, r766, r774);
        double r776 = r772 / r775;
        double r777 = r770 * r776;
        double r778 = sqrt(r777);
        double r779 = r769 * r778;
        double r780 = 1.0;
        double r781 = sqrt(r780);
        double r782 = r781 * r774;
        double r783 = r782 + r766;
        double r784 = r770 * r783;
        double r785 = sqrt(r784);
        double r786 = r769 * r785;
        double r787 = r768 ? r779 : r786;
        return r787;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.9
Target33.8
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -5.904936880911102e+122

    1. Initial program 62.2

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

    3. Applied flip-+62.2

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

    4. Simplified47.7

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

    5. Simplified31.7

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

    if -5.904936880911102e+122 < re

    1. Initial program 34.6

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

    3. Applied *-un-lft-identity34.6

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

    4. Applied sqrt-prod34.6

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

    5. Simplified8.3

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.90493688091110228 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{fma}\left(-1, re, \mathsf{hypot}\left(re, im\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))))