Average Error: 37.6 → 30.5
Time: 25.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\

\mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\
\;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r30441209 = 0.5;
        double r30441210 = 2.0;
        double r30441211 = re;
        double r30441212 = r30441211 * r30441211;
        double r30441213 = im;
        double r30441214 = r30441213 * r30441213;
        double r30441215 = r30441212 + r30441214;
        double r30441216 = sqrt(r30441215);
        double r30441217 = r30441216 + r30441211;
        double r30441218 = r30441210 * r30441217;
        double r30441219 = sqrt(r30441218);
        double r30441220 = r30441209 * r30441219;
        return r30441220;
}
↓
double f(double re, double im) {
        double r30441221 = im;
        double r30441222 = -1.8776884943596414e-117;
        bool r30441223 = r30441221 <= r30441222;
        double r30441224 = 2.0;
        double r30441225 = re;
        double r30441226 = r30441221 * r30441221;
        double r30441227 = r30441225 * r30441225;
        double r30441228 = r30441226 + r30441227;
        double r30441229 = cbrt(r30441228);
        double r30441230 = sqrt(r30441229);
        double r30441231 = sqrt(r30441230);
        double r30441232 = sqrt(r30441228);
        double r30441233 = sqrt(r30441232);
        double r30441234 = cbrt(r30441229);
        double r30441235 = r30441234 * r30441234;
        double r30441236 = r30441235 * r30441234;
        double r30441237 = r30441236 * r30441229;
        double r30441238 = sqrt(r30441237);
        double r30441239 = sqrt(r30441238);
        double r30441240 = r30441233 * r30441239;
        double r30441241 = r30441231 * r30441240;
        double r30441242 = r30441225 + r30441241;
        double r30441243 = r30441224 * r30441242;
        double r30441244 = sqrt(r30441243);
        double r30441245 = 0.5;
        double r30441246 = r30441244 * r30441245;
        double r30441247 = 1.2794987088213298e-198;
        bool r30441248 = r30441221 <= r30441247;
        double r30441249 = r30441225 + r30441225;
        double r30441250 = r30441249 * r30441224;
        double r30441251 = sqrt(r30441250);
        double r30441252 = r30441251 * r30441245;
        double r30441253 = r30441225 + r30441221;
        double r30441254 = r30441253 * r30441224;
        double r30441255 = sqrt(r30441254);
        double r30441256 = r30441255 * r30441245;
        double r30441257 = r30441248 ? r30441252 : r30441256;
        double r30441258 = r30441223 ? r30441246 : r30441257;
        return r30441258;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
Herbie30.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if im < -1.8776884943596414e-117

    1. Initial program 36.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Applied sqrt-prod36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    7. Applied sqrt-prod36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    8. Applied sqrt-prod36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right)} + re\right)}\]

    9. Applied associate-*r*36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt36.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    if -1.8776884943596414e-117 < im < 1.2794987088213298e-198

    1. Initial program 40.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt40.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Applied sqrt-prod41.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    5. Taylor expanded around inf 33.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]

    if 1.2794987088213298e-198 < im

    1. Initial program 37.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt37.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Applied sqrt-prod37.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    5. Taylor expanded around 0 23.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))