Average Error: 38.2 → 26.1
Time: 4.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.440559815511444601575174666981692354636 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 3.570900530608078795984829264447744999595 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.440559815511444601575174666981692354636 \cdot 10^{-295}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 3.570900530608078795984829264447744999595 \cdot 10^{107}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\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 r179053 = 0.5;
        double r179054 = 2.0;
        double r179055 = re;
        double r179056 = r179055 * r179055;
        double r179057 = im;
        double r179058 = r179057 * r179057;
        double r179059 = r179056 + r179058;
        double r179060 = sqrt(r179059);
        double r179061 = r179060 + r179055;
        double r179062 = r179054 * r179061;
        double r179063 = sqrt(r179062);
        double r179064 = r179053 * r179063;
        return r179064;
}
↓
double f(double re, double im) {
        double r179065 = re;
        double r179066 = -7.440559815511445e-295;
        bool r179067 = r179065 <= r179066;
        double r179068 = 0.5;
        double r179069 = 2.0;
        double r179070 = im;
        double r179071 = 2.0;
        double r179072 = pow(r179070, r179071);
        double r179073 = r179065 * r179065;
        double r179074 = r179070 * r179070;
        double r179075 = r179073 + r179074;
        double r179076 = sqrt(r179075);
        double r179077 = r179076 - r179065;
        double r179078 = r179072 / r179077;
        double r179079 = r179069 * r179078;
        double r179080 = sqrt(r179079);
        double r179081 = r179068 * r179080;
        double r179082 = 3.570900530608079e+107;
        bool r179083 = r179065 <= r179082;
        double r179084 = r179076 + r179065;
        double r179085 = r179069 * r179084;
        double r179086 = sqrt(r179085);
        double r179087 = r179068 * r179086;
        double r179088 = r179071 * r179065;
        double r179089 = r179069 * r179088;
        double r179090 = sqrt(r179089);
        double r179091 = r179068 * r179090;
        double r179092 = r179083 ? r179087 : r179091;
        double r179093 = r179067 ? r179081 : r179092;
        return r179093;
}

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
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -7.440559815511445e-295

    1. Initial program 45.7

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

    3. Applied flip-+45.5

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

    4. Simplified35.2

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

    if -7.440559815511445e-295 < re < 3.570900530608079e+107

    1. Initial program 21.1

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

    if 3.570900530608079e+107 < re

    1. Initial program 53.2

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.440559815511444601575174666981692354636 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 3.570900530608078795984829264447744999595 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))))