Average Error: 38.2 → 22.1
Time: 10.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

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

\end{array}
double f(double re, double im) {
        double r306136 = 0.5;
        double r306137 = 2.0;
        double r306138 = re;
        double r306139 = r306138 * r306138;
        double r306140 = im;
        double r306141 = r306140 * r306140;
        double r306142 = r306139 + r306141;
        double r306143 = sqrt(r306142);
        double r306144 = r306143 + r306138;
        double r306145 = r306137 * r306144;
        double r306146 = sqrt(r306145);
        double r306147 = r306136 * r306146;
        return r306147;
}
↓
double f(double re, double im) {
        double r306148 = re;
        double r306149 = -1.3292486390963865e+154;
        bool r306150 = r306148 <= r306149;
        double r306151 = im;
        double r306152 = r306151 * r306151;
        double r306153 = 2.0;
        double r306154 = r306152 * r306153;
        double r306155 = sqrt(r306154);
        double r306156 = -r306148;
        double r306157 = r306156 - r306148;
        double r306158 = sqrt(r306157);
        double r306159 = r306155 / r306158;
        double r306160 = 0.5;
        double r306161 = r306159 * r306160;
        double r306162 = 9.989819735270073e-90;
        bool r306163 = r306148 <= r306162;
        double r306164 = sqrt(r306153);
        double r306165 = fabs(r306151);
        double r306166 = r306164 * r306165;
        double r306167 = r306148 * r306148;
        double r306168 = r306167 + r306152;
        double r306169 = sqrt(r306168);
        double r306170 = r306169 - r306148;
        double r306171 = sqrt(r306170);
        double r306172 = r306166 / r306171;
        double r306173 = r306160 * r306172;
        double r306174 = 2.0;
        double r306175 = r306174 * r306148;
        double r306176 = r306153 * r306175;
        double r306177 = sqrt(r306176);
        double r306178 = r306160 * r306177;
        double r306179 = r306163 ? r306173 : r306178;
        double r306180 = r306150 ? r306161 : r306179;
        return r306180;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.2
Target33.2
Herbie22.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -1.3292486390963865e+154

    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. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified51.3

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

    7. Taylor expanded around -inf 21.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}{\sqrt{\color{blue}{-1 \cdot re} - re}}\]

    8. Simplified21.3

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

    if -1.3292486390963865e+154 < re < 9.989819735270073e-90

    1. Initial program 35.2

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

    3. Applied flip-+37.3

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

    4. Applied associate-*r/37.3

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

    5. Applied sqrt-div37.6

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

    6. Simplified30.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod30.5

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

    9. Simplified23.8

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

    if 9.989819735270073e-90 < re

    1. Initial program 33.8

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

    3. Applied add-cube-cbrt34.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Taylor expanded around inf 19.7

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))