math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 2.0 → 0.8

Time: 18.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -0.059112548828125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

C

double f(double re, double im) {
        double r1134554 = 0.5;
        double r1134555 = /* ERROR: no posit support in C */;
        double r1134556 = 2.0;
        double r1134557 = /* ERROR: no posit support in C */;
        double r1134558 = re;
        double r1134559 = r1134558 * r1134558;
        double r1134560 = im;
        double r1134561 = r1134560 * r1134560;
        double r1134562 = r1134559 + r1134561;
        double r1134563 = sqrt(r1134562);
        double r1134564 = r1134563 - r1134558;
        double r1134565 = r1134557 * r1134564;
        double r1134566 = sqrt(r1134565);
        double r1134567 = r1134555 * r1134566;
        return r1134567;
}

↓

double f(double re, double im) {
        double r1134568 = re;
        double r1134569 = -0.059112548828125;
        bool r1134570 = r1134568 <= r1134569;
        double r1134571 = 0.5;
        double r1134572 = 2.0;
        double r1134573 = r1134568 * r1134568;
        double r1134574 = im;
        double r1134575 = r1134574 * r1134574;
        double r1134576 = r1134573 + r1134575;
        double r1134577 = sqrt(r1134576);
        double r1134578 = r1134577 - r1134568;
        double r1134579 = r1134572 * r1134578;
        double r1134580 = sqrt(r1134579);
        double r1134581 = r1134571 * r1134580;
        double r1134582 = r1134575 + r1134573;
        double r1134583 = sqrt(r1134582);
        double r1134584 = r1134583 + r1134568;
        double r1134585 = r1134572 / r1134584;
        double r1134586 = 1.0;
        double r1134587 = r1134586 * r1134575;
        double r1134588 = r1134585 * r1134587;
        double r1134589 = sqrt(r1134588);
        double r1134590 = r1134589 * r1134571;
        double r1134591 = r1134570 ? r1134581 : r1134590;
        return r1134591;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 2 regimes

2. if re < -0.059112548828125

   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 -0.059112548828125 < re

   1. Initial program 3.1

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

   3. Applied p16-flip-- 2.9

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

   5. Applied difference-of-squares 3.1

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

   7. Applied p16-flip-- 2.9

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

   8. Applied associate-*r/ 3.5

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

   9. Applied associate-/l/ 3.5

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

   10. Simplified 1.8

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

   11. Simplified 0.9

       \[\leadsto \color{blue}{\left(\sqrt{\left(\left(\frac{\left(2.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(im \cdot im\right)}{\left(re \cdot re\right)}\right)}\right)}{re}\right)}\right) \cdot \left(\left(1.0\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.059112548828125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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)))))