Average Error: 2.0 → 2.0
Time: 18.8s
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 r1093505 = 0.5;
        double r1093506 = /* ERROR: no posit support in C */;
        double r1093507 = 2.0;
        double r1093508 = /* ERROR: no posit support in C */;
        double r1093509 = re;
        double r1093510 = r1093509 * r1093509;
        double r1093511 = im;
        double r1093512 = r1093511 * r1093511;
        double r1093513 = r1093510 + r1093512;
        double r1093514 = sqrt(r1093513);
        double r1093515 = r1093514 - r1093509;
        double r1093516 = r1093508 * r1093515;
        double r1093517 = sqrt(r1093516);
        double r1093518 = r1093506 * r1093517;
        return r1093518;
}
↓
double f(double re, double im) {
        double r1093519 = 0.5;
        double r1093520 = 2.0;
        double r1093521 = re;
        double r1093522 = r1093521 * r1093521;
        double r1093523 = im;
        double r1093524 = r1093523 * r1093523;
        double r1093525 = r1093522 + r1093524;
        double r1093526 = sqrt(r1093525);
        double r1093527 = r1093526 - r1093521;
        double r1093528 = r1093520 * r1093527;
        double r1093529 = sqrt(r1093528);
        double r1093530 = r1093519 * r1093529;
        return r1093530;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.0

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

  2. Final simplification2.0

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

Reproduce

herbie shell --seed 2019124 +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)))))