Average Error: 2.1 → 2.1
Time: 17.8s
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 r870697 = 0.5;
        double r870698 = /* ERROR: no posit support in C */;
        double r870699 = 2.0;
        double r870700 = /* ERROR: no posit support in C */;
        double r870701 = re;
        double r870702 = r870701 * r870701;
        double r870703 = im;
        double r870704 = r870703 * r870703;
        double r870705 = r870702 + r870704;
        double r870706 = sqrt(r870705);
        double r870707 = r870706 - r870701;
        double r870708 = r870700 * r870707;
        double r870709 = sqrt(r870708);
        double r870710 = r870698 * r870709;
        return r870710;
}
↓
double f(double re, double im) {
        double r870711 = 0.5;
        double r870712 = 2.0;
        double r870713 = re;
        double r870714 = r870713 * r870713;
        double r870715 = im;
        double r870716 = r870715 * r870715;
        double r870717 = r870714 + r870716;
        double r870718 = sqrt(r870717);
        double r870719 = r870718 - r870713;
        double r870720 = r870712 * r870719;
        double r870721 = sqrt(r870720);
        double r870722 = r870711 * r870721;
        return r870722;
}

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 +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)))))