Average Error: 2.1 → 0.7
Time: 10.1s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 0.00089263916015625:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le 0.00089263916015625:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\end{array}
double f(double re, double im) {
        double r229511 = 0.5;
        double r229512 = /* ERROR: no posit support in C */;
        double r229513 = 2.0;
        double r229514 = /* ERROR: no posit support in C */;
        double r229515 = re;
        double r229516 = r229515 * r229515;
        double r229517 = im;
        double r229518 = r229517 * r229517;
        double r229519 = r229516 + r229518;
        double r229520 = sqrt(r229519);
        double r229521 = r229520 - r229515;
        double r229522 = r229514 * r229521;
        double r229523 = sqrt(r229522);
        double r229524 = r229512 * r229523;
        return r229524;
}
↓
double f(double re, double im) {
        double r229525 = re;
        double r229526 = 0.00089263916015625;
        bool r229527 = r229525 <= r229526;
        double r229528 = 0.5;
        double r229529 = 2.0;
        double r229530 = r229525 * r229525;
        double r229531 = im;
        double r229532 = r229531 * r229531;
        double r229533 = r229530 + r229532;
        double r229534 = sqrt(r229533);
        double r229535 = r229534 - r229525;
        double r229536 = r229529 * r229535;
        double r229537 = sqrt(r229536);
        double r229538 = r229528 * r229537;
        double r229539 = r229534 + r229525;
        double r229540 = r229539 / r229531;
        double r229541 = r229531 / r229540;
        double r229542 = r229529 * r229541;
        double r229543 = sqrt(r229542);
        double r229544 = r229528 * r229543;
        double r229545 = r229527 ? r229538 : r229544;
        return r229545;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < 0.00089263916015625

    1. Initial program 0.7

      \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]

    if 0.00089263916015625 < re

    1. Initial program 3.5

      \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
    2. Using strategy rm

    3. Applied p16-flip--3.2

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)}\right)}\right)\]

    4. Simplified0.9

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(im \cdot im\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-/l*0.8

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{im}{\left(\frac{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}{im}\right)}\right)}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 0.00089263916015625:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (-.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))