Average Error: 2.1 → 2.1
Time: 17.2s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
\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)
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
double f(double re, double im) {
        double r863323 = 0.5;
        double r863324 = /* ERROR: no posit support in C */;
        double r863325 = 2.0;
        double r863326 = /* ERROR: no posit support in C */;
        double r863327 = re;
        double r863328 = r863327 * r863327;
        double r863329 = im;
        double r863330 = r863329 * r863329;
        double r863331 = r863328 + r863330;
        double r863332 = sqrt(r863331);
        double r863333 = r863332 - r863327;
        double r863334 = r863326 * r863333;
        double r863335 = sqrt(r863334);
        double r863336 = r863324 * r863335;
        return r863336;
}
↓
double f(double re, double im) {
        double r863337 = 0.5;
        double r863338 = 2.0;
        double r863339 = re;
        double r863340 = r863339 * r863339;
        double r863341 = im;
        double r863342 = r863341 * r863341;
        double r863343 = r863340 + r863342;
        double r863344 = sqrt(r863343);
        double r863345 = r863344 - r863339;
        double r863346 = r863338 * r863345;
        double r863347 = sqrt(r863346);
        double r863348 = r863337 * r863347;
        return r863348;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

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

  2. Final simplification2.1

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

Reproduce

herbie shell --seed 2019120 
(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)))))