Average Error: 38.9 → 11.9
Time: 4.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.47512303500360515 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.47512303500360515 \cdot 10^{161}:\\
\;\;\;\;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(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r191138 = 0.5;
        double r191139 = 2.0;
        double r191140 = re;
        double r191141 = r191140 * r191140;
        double r191142 = im;
        double r191143 = r191142 * r191142;
        double r191144 = r191141 + r191143;
        double r191145 = sqrt(r191144);
        double r191146 = r191145 + r191140;
        double r191147 = r191139 * r191146;
        double r191148 = sqrt(r191147);
        double r191149 = r191138 * r191148;
        return r191149;
}
↓
double f(double re, double im) {
        double r191150 = re;
        double r191151 = -8.475123035003605e+161;
        bool r191152 = r191150 <= r191151;
        double r191153 = 0.5;
        double r191154 = 2.0;
        double r191155 = 0.0;
        double r191156 = im;
        double r191157 = 2.0;
        double r191158 = pow(r191156, r191157);
        double r191159 = r191155 + r191158;
        double r191160 = hypot(r191150, r191156);
        double r191161 = r191160 - r191150;
        double r191162 = r191159 / r191161;
        double r191163 = r191154 * r191162;
        double r191164 = sqrt(r191163);
        double r191165 = r191153 * r191164;
        double r191166 = 1.0;
        double r191167 = sqrt(r191166);
        double r191168 = r191167 * r191160;
        double r191169 = r191168 + r191150;
        double r191170 = r191154 * r191169;
        double r191171 = sqrt(r191170);
        double r191172 = r191153 * r191171;
        double r191173 = r191152 ? r191165 : r191172;
        return r191173;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.9
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 < -8.475123035003605e+161

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Simplified50.5

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

    5. Simplified30.2

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

    if -8.475123035003605e+161 < re

    1. Initial program 35.5

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

    3. Applied *-un-lft-identity35.5

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

    4. Applied sqrt-prod35.5

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

    5. Simplified9.4

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.47512303500360515 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))))