Average Error: 38.8 → 26.6
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0699181930618242 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im - 0}{-1 \cdot re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \le 1.68301247060889316 \cdot 10^{70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0699181930618242 \cdot 10^{-303}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im - 0}{-1 \cdot re + \sqrt{re \cdot re + im \cdot im}}}\\

\mathbf{elif}\;re \le 1.68301247060889316 \cdot 10^{70}:\\
\;\;\;\;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 r146977 = 0.5;
        double r146978 = 2.0;
        double r146979 = re;
        double r146980 = r146979 * r146979;
        double r146981 = im;
        double r146982 = r146981 * r146981;
        double r146983 = r146980 + r146982;
        double r146984 = sqrt(r146983);
        double r146985 = r146984 + r146979;
        double r146986 = r146978 * r146985;
        double r146987 = sqrt(r146986);
        double r146988 = r146977 * r146987;
        return r146988;
}
↓
double f(double re, double im) {
        double r146989 = re;
        double r146990 = -1.0699181930618242e-303;
        bool r146991 = r146989 <= r146990;
        double r146992 = 0.5;
        double r146993 = 2.0;
        double r146994 = im;
        double r146995 = r146994 * r146994;
        double r146996 = 0.0;
        double r146997 = r146995 - r146996;
        double r146998 = -1.0;
        double r146999 = r146998 * r146989;
        double r147000 = r146989 * r146989;
        double r147001 = r147000 + r146995;
        double r147002 = sqrt(r147001);
        double r147003 = r146999 + r147002;
        double r147004 = r146997 / r147003;
        double r147005 = r146993 * r147004;
        double r147006 = sqrt(r147005);
        double r147007 = r146992 * r147006;
        double r147008 = 1.6830124706088932e+70;
        bool r147009 = r146989 <= r147008;
        double r147010 = r147002 + r146989;
        double r147011 = r146993 * r147010;
        double r147012 = sqrt(r147011);
        double r147013 = r146992 * r147012;
        double r147014 = 2.0;
        double r147015 = r147014 * r146989;
        double r147016 = r146993 * r147015;
        double r147017 = sqrt(r147016);
        double r147018 = r146992 * r147017;
        double r147019 = r147009 ? r147013 : r147018;
        double r147020 = r146991 ? r147007 : r147019;
        return r147020;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.6
Herbie26.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -1.0699181930618242e-303

    1. Initial program 45.4

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

    3. Applied add-cube-cbrt46.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+46.5

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

    6. Simplified35.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im - 0}}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re}}\]

    7. Simplified34.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im - 0}{\color{blue}{-1 \cdot re + \sqrt{re \cdot re + im \cdot im}}}}\]

    if -1.0699181930618242e-303 < re < 1.6830124706088932e+70

    1. Initial program 22.6

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

    if 1.6830124706088932e+70 < re

    1. Initial program 48.1

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

    3. Applied add-cube-cbrt48.3

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

    4. Taylor expanded around inf 11.7

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

  4. Final simplification26.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0699181930618242 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im - 0}{-1 \cdot re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \le 1.68301247060889316 \cdot 10^{70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))))