Average Error: 39.1 → 22.9
Time: 3.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3657760091869803 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\ \mathbf{elif}\;re \le 1.5366724406791122 \cdot 10^{106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.3657760091869803 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\

\mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\

\mathbf{elif}\;re \le 1.5366724406791122 \cdot 10^{106}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r130114 = 0.5;
        double r130115 = 2.0;
        double r130116 = re;
        double r130117 = r130116 * r130116;
        double r130118 = im;
        double r130119 = r130118 * r130118;
        double r130120 = r130117 + r130119;
        double r130121 = sqrt(r130120);
        double r130122 = r130121 + r130116;
        double r130123 = r130115 * r130122;
        double r130124 = sqrt(r130123);
        double r130125 = r130114 * r130124;
        return r130125;
}
↓
double f(double re, double im) {
        double r130126 = re;
        double r130127 = -1.3657760091869803e+154;
        bool r130128 = r130126 <= r130127;
        double r130129 = 0.5;
        double r130130 = 2.0;
        double r130131 = im;
        double r130132 = -2.0;
        double r130133 = r130132 * r130126;
        double r130134 = r130131 / r130133;
        double r130135 = r130131 * r130134;
        double r130136 = r130130 * r130135;
        double r130137 = sqrt(r130136);
        double r130138 = r130129 * r130137;
        double r130139 = -4.540545293771823e-275;
        bool r130140 = r130126 <= r130139;
        double r130141 = r130131 * r130131;
        double r130142 = r130130 * r130141;
        double r130143 = sqrt(r130142);
        double r130144 = r130126 * r130126;
        double r130145 = r130144 + r130141;
        double r130146 = sqrt(r130145);
        double r130147 = r130146 - r130126;
        double r130148 = sqrt(r130147);
        double r130149 = r130143 / r130148;
        double r130150 = r130129 * r130149;
        double r130151 = 1.2766285812733717e-281;
        bool r130152 = r130126 <= r130151;
        double r130153 = r130131 - r130126;
        double r130154 = r130131 / r130153;
        double r130155 = r130131 * r130154;
        double r130156 = r130130 * r130155;
        double r130157 = sqrt(r130156);
        double r130158 = r130129 * r130157;
        double r130159 = 1.5366724406791122e+106;
        bool r130160 = r130126 <= r130159;
        double r130161 = sqrt(r130146);
        double r130162 = r130161 * r130161;
        double r130163 = r130162 + r130126;
        double r130164 = r130130 * r130163;
        double r130165 = sqrt(r130164);
        double r130166 = r130129 * r130165;
        double r130167 = 2.0;
        double r130168 = r130167 * r130126;
        double r130169 = r130130 * r130168;
        double r130170 = sqrt(r130169);
        double r130171 = r130129 * r130170;
        double r130172 = r130160 ? r130166 : r130171;
        double r130173 = r130152 ? r130158 : r130172;
        double r130174 = r130140 ? r130150 : r130173;
        double r130175 = r130128 ? r130138 : r130174;
        return r130175;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target34.1
Herbie22.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.3657760091869803e+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. Simplified50.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity50.6

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

    7. Applied times-frac50.2

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

    8. Simplified50.2

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

    9. Taylor expanded around -inf 22.8

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

    if -1.3657760091869803e+154 < re < -4.540545293771823e-275

    1. Initial program 41.2

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

    3. Applied flip-+41.0

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

    4. Simplified31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-*r/31.4

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

    7. Applied sqrt-div30.0

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

    if -4.540545293771823e-275 < re < 1.2766285812733717e-281

    1. Initial program 32.3

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

    3. Applied flip-+32.4

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

    4. Simplified32.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity32.4

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

    7. Applied times-frac31.8

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

    8. Simplified31.8

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

    9. Taylor expanded around 0 33.4

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

    if 1.2766285812733717e-281 < re < 1.5366724406791122e+106

    1. Initial program 20.7

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

    3. Applied add-sqr-sqrt20.7

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

    4. Applied sqrt-prod20.8

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

    if 1.5366724406791122e+106 < re

    1. Initial program 52.7

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

    3. Applied flip-+63.5

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

    4. Simplified62.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity62.6

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

    7. Applied times-frac62.2

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

    8. Simplified62.2

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

    9. Taylor expanded around 0 8.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification22.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3657760091869803 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\ \mathbf{elif}\;re \le 1.5366724406791122 \cdot 10^{106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))