Average Error: 38.9 → 27.4
Time: 13.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.797060425996081289789292529027987197048 \cdot 10^{81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.796065771880538982336997204574724695612 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)}\right) - re\right)}\\ \mathbf{elif}\;re \le 7.84333861213225274022933802875203231721 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.797060425996081289789292529027987197048 \cdot 10^{81}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -1.796065771880538982336997204574724695612 \cdot 10^{-203}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)}\right) - re\right)}\\

\mathbf{elif}\;re \le 7.84333861213225274022933802875203231721 \cdot 10^{-202}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + \sqrt{re \cdot re + im \cdot im}}}\\

\end{array}
double f(double re, double im) {
        double r23061 = 0.5;
        double r23062 = 2.0;
        double r23063 = re;
        double r23064 = r23063 * r23063;
        double r23065 = im;
        double r23066 = r23065 * r23065;
        double r23067 = r23064 + r23066;
        double r23068 = sqrt(r23067);
        double r23069 = r23068 - r23063;
        double r23070 = r23062 * r23069;
        double r23071 = sqrt(r23070);
        double r23072 = r23061 * r23071;
        return r23072;
}
↓
double f(double re, double im) {
        double r23073 = re;
        double r23074 = -2.7970604259960813e+81;
        bool r23075 = r23073 <= r23074;
        double r23076 = 0.5;
        double r23077 = 2.0;
        double r23078 = -2.0;
        double r23079 = r23078 * r23073;
        double r23080 = r23077 * r23079;
        double r23081 = sqrt(r23080);
        double r23082 = r23076 * r23081;
        double r23083 = -1.796065771880539e-203;
        bool r23084 = r23073 <= r23083;
        double r23085 = r23073 * r23073;
        double r23086 = im;
        double r23087 = r23086 * r23086;
        double r23088 = r23085 + r23087;
        double r23089 = sqrt(r23088);
        double r23090 = sqrt(r23089);
        double r23091 = sqrt(r23090);
        double r23092 = 3.0;
        double r23093 = 2.0;
        double r23094 = r23092 / r23093;
        double r23095 = pow(r23091, r23094);
        double r23096 = r23095 * r23095;
        double r23097 = r23091 * r23096;
        double r23098 = r23097 - r23073;
        double r23099 = r23077 * r23098;
        double r23100 = sqrt(r23099);
        double r23101 = r23076 * r23100;
        double r23102 = 7.843338612132253e-202;
        bool r23103 = r23073 <= r23102;
        double r23104 = r23086 - r23073;
        double r23105 = r23077 * r23104;
        double r23106 = sqrt(r23105);
        double r23107 = r23076 * r23106;
        double r23108 = pow(r23086, r23093);
        double r23109 = r23073 + r23089;
        double r23110 = r23108 / r23109;
        double r23111 = r23077 * r23110;
        double r23112 = sqrt(r23111);
        double r23113 = r23076 * r23112;
        double r23114 = r23103 ? r23107 : r23113;
        double r23115 = r23084 ? r23101 : r23114;
        double r23116 = r23075 ? r23082 : r23115;
        return r23116;
}

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 4 regimes

  2. if re < -2.7970604259960813e+81

    1. Initial program 48.9

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

    2. Taylor expanded around -inf 11.3

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

    if -2.7970604259960813e+81 < re < -1.796065771880539e-203

    1. Initial program 18.5

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

    3. Applied add-sqr-sqrt18.5

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

    4. Applied sqrt-prod18.5

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

    6. Applied add-sqr-sqrt18.5

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

    7. Applied sqrt-prod18.5

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

    8. Applied sqrt-prod18.6

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

    9. Applied associate-*l*18.6

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

    10. Simplified18.7

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

    12. Applied sqr-pow18.7

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

    if -1.796065771880539e-203 < re < 7.843338612132253e-202

    1. Initial program 31.1

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

    2. Taylor expanded around 0 33.8

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

    if 7.843338612132253e-202 < re

    1. Initial program 49.1

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

    3. Applied flip--49.1

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

    4. Simplified37.0

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

    5. Simplified37.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{re + \sqrt{re \cdot re + im \cdot im}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.797060425996081289789292529027987197048 \cdot 10^{81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.796065771880538982336997204574724695612 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\frac{3}{2}\right)}\right) - re\right)}\\ \mathbf{elif}\;re \le 7.84333861213225274022933802875203231721 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))))