Average Error: 2.0 → 0.8
Time: 10.1s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -0.014678955078125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]
\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 -0.014678955078125:\\
\;\;\;\;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 r844335 = 0.5;
        double r844336 = /* ERROR: no posit support in C */;
        double r844337 = 2.0;
        double r844338 = /* ERROR: no posit support in C */;
        double r844339 = re;
        double r844340 = r844339 * r844339;
        double r844341 = im;
        double r844342 = r844341 * r844341;
        double r844343 = r844340 + r844342;
        double r844344 = sqrt(r844343);
        double r844345 = r844344 - r844339;
        double r844346 = r844338 * r844345;
        double r844347 = sqrt(r844346);
        double r844348 = r844336 * r844347;
        return r844348;
}
↓
double f(double re, double im) {
        double r844349 = re;
        double r844350 = -0.014678955078125;
        bool r844351 = r844349 <= r844350;
        double r844352 = 0.5;
        double r844353 = 2.0;
        double r844354 = r844349 * r844349;
        double r844355 = im;
        double r844356 = r844355 * r844355;
        double r844357 = r844354 + r844356;
        double r844358 = sqrt(r844357);
        double r844359 = r844358 - r844349;
        double r844360 = r844353 * r844359;
        double r844361 = sqrt(r844360);
        double r844362 = r844352 * r844361;
        double r844363 = r844358 + r844349;
        double r844364 = r844363 / r844355;
        double r844365 = r844355 / r844364;
        double r844366 = r844353 * r844365;
        double r844367 = sqrt(r844366);
        double r844368 = r844352 * r844367;
        double r844369 = r844351 ? r844362 : r844368;
        return r844369;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < -0.014678955078125

    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 -0.014678955078125 < re

    1. Initial program 3.3

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

    3. Applied p16-flip--3.0

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

    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 simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.014678955078125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))))