Average Error: 39.2 → 19.2
Time: 14.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.178594449777199163791033377764152427439 \cdot 10^{109}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;re \le -2.831188104063089421579428004178246992819 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.819440950287977262527658386462127746153 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{2 \cdot re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.178594449777199163791033377764152427439 \cdot 10^{109}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\

\mathbf{elif}\;re \le -2.831188104063089421579428004178246992819 \cdot 10^{-307}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le 1.819440950287977262527658386462127746153 \cdot 10^{151}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\

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

\end{array}
double f(double re, double im) {
        double r27519 = 0.5;
        double r27520 = 2.0;
        double r27521 = re;
        double r27522 = r27521 * r27521;
        double r27523 = im;
        double r27524 = r27523 * r27523;
        double r27525 = r27522 + r27524;
        double r27526 = sqrt(r27525);
        double r27527 = r27526 - r27521;
        double r27528 = r27520 * r27527;
        double r27529 = sqrt(r27528);
        double r27530 = r27519 * r27529;
        return r27530;
}
↓
double f(double re, double im) {
        double r27531 = re;
        double r27532 = -1.1785944497771992e+109;
        bool r27533 = r27531 <= r27532;
        double r27534 = 0.5;
        double r27535 = 2.0;
        double r27536 = -r27531;
        double r27537 = r27536 - r27531;
        double r27538 = r27535 * r27537;
        double r27539 = sqrt(r27538);
        double r27540 = r27534 * r27539;
        double r27541 = -2.8311881040630894e-307;
        bool r27542 = r27531 <= r27541;
        double r27543 = r27531 * r27531;
        double r27544 = im;
        double r27545 = r27544 * r27544;
        double r27546 = r27543 + r27545;
        double r27547 = sqrt(r27546);
        double r27548 = sqrt(r27547);
        double r27549 = r27548 * r27548;
        double r27550 = r27549 - r27531;
        double r27551 = r27535 * r27550;
        double r27552 = sqrt(r27551);
        double r27553 = r27534 * r27552;
        double r27554 = 1.8194409502879773e+151;
        bool r27555 = r27531 <= r27554;
        double r27556 = sqrt(r27535);
        double r27557 = r27547 + r27531;
        double r27558 = sqrt(r27557);
        double r27559 = fabs(r27544);
        double r27560 = r27558 / r27559;
        double r27561 = r27556 / r27560;
        double r27562 = r27534 * r27561;
        double r27563 = r27545 * r27535;
        double r27564 = sqrt(r27563);
        double r27565 = 2.0;
        double r27566 = r27565 * r27531;
        double r27567 = sqrt(r27566);
        double r27568 = r27564 / r27567;
        double r27569 = r27534 * r27568;
        double r27570 = r27555 ? r27562 : r27569;
        double r27571 = r27542 ? r27553 : r27570;
        double r27572 = r27533 ? r27540 : r27571;
        return r27572;
}

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 < -1.1785944497771992e+109

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 11.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

    3. Simplified11.4

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

    if -1.1785944497771992e+109 < re < -2.8311881040630894e-307

    1. Initial program 21.6

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

    3. Applied add-sqr-sqrt21.6

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

    4. Applied sqrt-prod21.7

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

    if -2.8311881040630894e-307 < re < 1.8194409502879773e+151

    1. Initial program 40.8

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

    3. Applied flip--40.6

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

    4. Applied associate-*r/40.6

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

    5. Applied sqrt-div40.7

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

    6. Simplified30.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod30.3

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

    9. Applied associate-/l*30.4

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

    10. Simplified20.2

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

    if 1.8194409502879773e+151 < re

    1. Initial program 63.8

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

    3. Applied flip--63.8

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

    4. Applied associate-*r/63.8

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

    5. Applied sqrt-div63.8

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

    6. Simplified50.0

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

    7. Taylor expanded around inf 18.9

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{2 \cdot re}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.178594449777199163791033377764152427439 \cdot 10^{109}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;re \le -2.831188104063089421579428004178246992819 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.819440950287977262527658386462127746153 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{2 \cdot re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))))