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

Average Error: 2.0 → 0.8

Time: 19.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le 0.0106658935546875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

C

double f(double re, double im) {
        double r1134529 = 0.5;
        double r1134530 = /* ERROR: no posit support in C */;
        double r1134531 = 2.0;
        double r1134532 = /* ERROR: no posit support in C */;
        double r1134533 = re;
        double r1134534 = r1134533 * r1134533;
        double r1134535 = im;
        double r1134536 = r1134535 * r1134535;
        double r1134537 = r1134534 + r1134536;
        double r1134538 = sqrt(r1134537);
        double r1134539 = r1134538 - r1134533;
        double r1134540 = r1134532 * r1134539;
        double r1134541 = sqrt(r1134540);
        double r1134542 = r1134530 * r1134541;
        return r1134542;
}

↓

double f(double re, double im) {
        double r1134543 = re;
        double r1134544 = 0.0106658935546875;
        bool r1134545 = r1134543 <= r1134544;
        double r1134546 = 0.5;
        double r1134547 = 2.0;
        double r1134548 = r1134543 * r1134543;
        double r1134549 = im;
        double r1134550 = r1134549 * r1134549;
        double r1134551 = r1134548 + r1134550;
        double r1134552 = sqrt(r1134551);
        double r1134553 = r1134552 - r1134543;
        double r1134554 = r1134547 * r1134553;
        double r1134555 = sqrt(r1134554);
        double r1134556 = r1134546 * r1134555;
        double r1134557 = r1134550 + r1134548;
        double r1134558 = sqrt(r1134557);
        double r1134559 = r1134558 + r1134543;
        double r1134560 = r1134547 / r1134559;
        double r1134561 = 1.0;
        double r1134562 = r1134561 * r1134550;
        double r1134563 = r1134560 * r1134562;
        double r1134564 = sqrt(r1134563);
        double r1134565 = r1134564 * r1134546;
        double r1134566 = r1134545 ? r1134556 : r1134565;
        return r1134566;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 2 regimes

2. if re < 0.0106658935546875

   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 0.0106658935546875 < 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-- 3.3

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

   5. Applied difference-of-squares 3.6

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

   7. Applied p16-flip-- 3.3

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

   8. Applied associate-*r/ 3.8

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

   9. Applied associate-/l/ 3.8

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

   10. Simplified 1.7

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

   11. Simplified 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 0.0106658935546875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))))