Average Error: 38.4 → 30.6
Time: 16.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 2.785635358716614812340651792624574520043 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;im \cdot im \le 7.885845283722595201168636922497800994676 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;im \cdot im \le 4.636456314439879455015808630226497398272 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.606608869031663108780796224948697957317 \cdot 10^{272}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \le 2.785635358716614812340651792624574520043 \cdot 10^{-267}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\mathbf{elif}\;im \cdot im \le 7.885845283722595201168636922497800994676 \cdot 10^{-207}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;im \cdot im \le 4.636456314439879455015808630226497398272 \cdot 10^{-190}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\mathbf{elif}\;im \cdot im \le 1.606608869031663108780796224948697957317 \cdot 10^{272}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

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

\end{array}
double f(double re, double im) {
        double r159119 = 0.5;
        double r159120 = 2.0;
        double r159121 = re;
        double r159122 = r159121 * r159121;
        double r159123 = im;
        double r159124 = r159123 * r159123;
        double r159125 = r159122 + r159124;
        double r159126 = sqrt(r159125);
        double r159127 = r159126 + r159121;
        double r159128 = r159120 * r159127;
        double r159129 = sqrt(r159128);
        double r159130 = r159119 * r159129;
        return r159130;
}
↓
double f(double re, double im) {
        double r159131 = im;
        double r159132 = r159131 * r159131;
        double r159133 = 2.785635358716615e-267;
        bool r159134 = r159132 <= r159133;
        double r159135 = 0.5;
        double r159136 = 2.0;
        double r159137 = re;
        double r159138 = r159137 + r159137;
        double r159139 = r159136 * r159138;
        double r159140 = sqrt(r159139);
        double r159141 = r159135 * r159140;
        double r159142 = 7.885845283722595e-207;
        bool r159143 = r159132 <= r159142;
        double r159144 = r159137 * r159137;
        double r159145 = r159144 + r159132;
        double r159146 = sqrt(r159145);
        double r159147 = r159146 - r159137;
        double r159148 = r159132 / r159147;
        double r159149 = r159136 * r159148;
        double r159150 = sqrt(r159149);
        double r159151 = r159135 * r159150;
        double r159152 = 4.636456314439879e-190;
        bool r159153 = r159132 <= r159152;
        double r159154 = 1.606608869031663e+272;
        bool r159155 = r159132 <= r159154;
        double r159156 = r159131 + r159137;
        double r159157 = r159136 * r159156;
        double r159158 = sqrt(r159157);
        double r159159 = r159135 * r159158;
        double r159160 = r159155 ? r159151 : r159159;
        double r159161 = r159153 ? r159141 : r159160;
        double r159162 = r159143 ? r159151 : r159161;
        double r159163 = r159134 ? r159141 : r159162;
        return r159163;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.5
Herbie30.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* im im) < 2.785635358716615e-267 or 7.885845283722595e-207 < (* im im) < 4.636456314439879e-190

    1. Initial program 41.4

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

    2. Taylor expanded around inf 36.4

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

    if 2.785635358716615e-267 < (* im im) < 7.885845283722595e-207 or 4.636456314439879e-190 < (* im im) < 1.606608869031663e+272

    1. Initial program 22.8

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

    3. Applied flip-+30.2

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

    4. Simplified22.9

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

    if 1.606608869031663e+272 < (* im im)

    1. Initial program 59.2

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

    3. Applied add-sqr-sqrt59.2

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

    4. Applied sqrt-prod59.2

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

    5. Taylor expanded around 0 36.1

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

  4. Final simplification30.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 2.785635358716614812340651792624574520043 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;im \cdot im \le 7.885845283722595201168636922497800994676 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;im \cdot im \le 4.636456314439879455015808630226497398272 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.606608869031663108780796224948697957317 \cdot 10^{272}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))))