Average Error: 38.8 → 13.1
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -2.047959521004422594885937246194657560109 \cdot 10^{153} \lor \neg \left(im \le -5.915675441341099358122284237091288586253 \cdot 10^{60} \lor \neg \left(im \le -645644762598500.5 \lor \neg \left(im \le -2.780954356842541639179543979500359083621 \cdot 10^{-41}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -2.047959521004422594885937246194657560109 \cdot 10^{153} \lor \neg \left(im \le -5.915675441341099358122284237091288586253 \cdot 10^{60} \lor \neg \left(im \le -645644762598500.5 \lor \neg \left(im \le -2.780954356842541639179543979500359083621 \cdot 10^{-41}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r18056 = 0.5;
        double r18057 = 2.0;
        double r18058 = re;
        double r18059 = r18058 * r18058;
        double r18060 = im;
        double r18061 = r18060 * r18060;
        double r18062 = r18059 + r18061;
        double r18063 = sqrt(r18062);
        double r18064 = r18063 - r18058;
        double r18065 = r18057 * r18064;
        double r18066 = sqrt(r18065);
        double r18067 = r18056 * r18066;
        return r18067;
}
↓
double f(double re, double im) {
        double r18068 = im;
        double r18069 = -2.0479595210044226e+153;
        bool r18070 = r18068 <= r18069;
        double r18071 = -5.9156754413410994e+60;
        bool r18072 = r18068 <= r18071;
        double r18073 = -645644762598500.5;
        bool r18074 = r18068 <= r18073;
        double r18075 = -2.7809543568425416e-41;
        bool r18076 = r18068 <= r18075;
        double r18077 = !r18076;
        bool r18078 = r18074 || r18077;
        double r18079 = !r18078;
        bool r18080 = r18072 || r18079;
        double r18081 = !r18080;
        bool r18082 = r18070 || r18081;
        double r18083 = 0.5;
        double r18084 = 2.0;
        double r18085 = re;
        double r18086 = hypot(r18085, r18068);
        double r18087 = r18086 - r18085;
        double r18088 = 0.0;
        double r18089 = r18087 + r18088;
        double r18090 = r18084 * r18089;
        double r18091 = sqrt(r18090);
        double r18092 = r18083 * r18091;
        double r18093 = 2.0;
        double r18094 = pow(r18068, r18093);
        double r18095 = r18094 + r18088;
        double r18096 = r18085 + r18086;
        double r18097 = r18095 / r18096;
        double r18098 = r18084 * r18097;
        double r18099 = sqrt(r18098);
        double r18100 = r18083 * r18099;
        double r18101 = r18082 ? r18092 : r18100;
        return r18101;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if im < -2.0479595210044226e+153 or -5.9156754413410994e+60 < im < -645644762598500.5 or -2.7809543568425416e-41 < im

    1. Initial program 41.5

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

    3. Applied add-cube-cbrt42.1

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

    4. Applied add-sqr-sqrt42.1

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

    5. Applied sqrt-prod42.2

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

    6. Applied prod-diff42.3

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

    7. Simplified15.8

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

    8. Simplified13.3

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

    if -2.0479595210044226e+153 < im < -5.9156754413410994e+60 or -645644762598500.5 < im < -2.7809543568425416e-41

    1. Initial program 18.9

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

    3. Applied flip--22.7

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

    4. Simplified18.6

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

    5. Simplified11.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.047959521004422594885937246194657560109 \cdot 10^{153} \lor \neg \left(im \le -5.915675441341099358122284237091288586253 \cdot 10^{60} \lor \neg \left(im \le -645644762598500.5 \lor \neg \left(im \le -2.780954356842541639179543979500359083621 \cdot 10^{-41}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))