Average Error: 38.6 → 25.7
Time: 35.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.553630577599801566658039576459712943161 \cdot 10^{-263}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.367500206157556418859171948354483295872 \cdot 10^{-222}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2}\\ \mathbf{elif}\;re \le 2.433220323045317356248333226770764594523 \cdot 10^{119}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.553630577599801566658039576459712943161 \cdot 10^{-263}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 1.367500206157556418859171948354483295872 \cdot 10^{-222}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2}\\

\mathbf{elif}\;re \le 2.433220323045317356248333226770764594523 \cdot 10^{119}:\\
\;\;\;\;\sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r7734052 = 0.5;
        double r7734053 = 2.0;
        double r7734054 = re;
        double r7734055 = r7734054 * r7734054;
        double r7734056 = im;
        double r7734057 = r7734056 * r7734056;
        double r7734058 = r7734055 + r7734057;
        double r7734059 = sqrt(r7734058);
        double r7734060 = r7734059 + r7734054;
        double r7734061 = r7734053 * r7734060;
        double r7734062 = sqrt(r7734061);
        double r7734063 = r7734052 * r7734062;
        return r7734063;
}
↓
double f(double re, double im) {
        double r7734064 = re;
        double r7734065 = -5.5536305775998016e-263;
        bool r7734066 = r7734064 <= r7734065;
        double r7734067 = im;
        double r7734068 = r7734067 * r7734067;
        double r7734069 = 2.0;
        double r7734070 = r7734068 * r7734069;
        double r7734071 = sqrt(r7734070);
        double r7734072 = r7734064 * r7734064;
        double r7734073 = r7734068 + r7734072;
        double r7734074 = sqrt(r7734073);
        double r7734075 = r7734074 - r7734064;
        double r7734076 = sqrt(r7734075);
        double r7734077 = r7734071 / r7734076;
        double r7734078 = 0.5;
        double r7734079 = r7734077 * r7734078;
        double r7734080 = 1.3675002061575564e-222;
        bool r7734081 = r7734064 <= r7734080;
        double r7734082 = r7734067 + r7734064;
        double r7734083 = r7734082 * r7734069;
        double r7734084 = sqrt(r7734083);
        double r7734085 = r7734078 * r7734084;
        double r7734086 = 2.4332203230453174e+119;
        bool r7734087 = r7734064 <= r7734086;
        double r7734088 = r7734064 + r7734074;
        double r7734089 = r7734069 * r7734088;
        double r7734090 = sqrt(r7734089);
        double r7734091 = r7734090 * r7734078;
        double r7734092 = r7734064 + r7734064;
        double r7734093 = r7734069 * r7734092;
        double r7734094 = sqrt(r7734093);
        double r7734095 = r7734094 * r7734078;
        double r7734096 = r7734087 ? r7734091 : r7734095;
        double r7734097 = r7734081 ? r7734085 : r7734096;
        double r7734098 = r7734066 ? r7734079 : r7734097;
        return r7734098;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -5.5536305775998016e-263

    1. Initial program 47.1

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

    3. Applied add-exp-log49.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+49.2

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

    6. Applied associate-*r/49.2

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

    7. Applied sqrt-div49.2

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

    8. Simplified36.2

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

    9. Simplified35.0

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

    if -5.5536305775998016e-263 < re < 1.3675002061575564e-222

    1. Initial program 31.9

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

    2. Taylor expanded around 0 31.6

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

    if 1.3675002061575564e-222 < re < 2.4332203230453174e+119

    1. Initial program 17.7

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

    if 2.4332203230453174e+119 < re

    1. Initial program 54.6

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

    2. Taylor expanded around inf 8.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.553630577599801566658039576459712943161 \cdot 10^{-263}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.367500206157556418859171948354483295872 \cdot 10^{-222}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2}\\ \mathbf{elif}\;re \le 2.433220323045317356248333226770764594523 \cdot 10^{119}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))