Average Error: 2.0 → 2.0
Time: 19.6s
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 r1342620 = 0.5;
        double r1342621 = /* ERROR: no posit support in C */;
        double r1342622 = 2.0;
        double r1342623 = /* ERROR: no posit support in C */;
        double r1342624 = re;
        double r1342625 = r1342624 * r1342624;
        double r1342626 = im;
        double r1342627 = r1342626 * r1342626;
        double r1342628 = r1342625 + r1342627;
        double r1342629 = sqrt(r1342628);
        double r1342630 = r1342629 - r1342624;
        double r1342631 = r1342623 * r1342630;
        double r1342632 = sqrt(r1342631);
        double r1342633 = r1342621 * r1342632;
        return r1342633;
}
↓
double f(double re, double im) {
        double r1342634 = 0.5;
        double r1342635 = 2.0;
        double r1342636 = re;
        double r1342637 = r1342636 * r1342636;
        double r1342638 = im;
        double r1342639 = r1342638 * r1342638;
        double r1342640 = r1342637 + r1342639;
        double r1342641 = sqrt(r1342640);
        double r1342642 = r1342641 - r1342636;
        double r1342643 = r1342635 * r1342642;
        double r1342644 = sqrt(r1342643);
        double r1342645 = r1342634 * r1342644;
        return r1342645;
}

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 
(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)))))