Average Error: 2.1 → 2.1
Time: 33.7s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]
\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{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}
double f(double re, double im) {
        double r1408897 = 0.5;
        double r1408898 = /* ERROR: no posit support in C */;
        double r1408899 = 2.0;
        double r1408900 = /* ERROR: no posit support in C */;
        double r1408901 = re;
        double r1408902 = r1408901 * r1408901;
        double r1408903 = im;
        double r1408904 = r1408903 * r1408903;
        double r1408905 = r1408902 + r1408904;
        double r1408906 = sqrt(r1408905);
        double r1408907 = r1408906 - r1408901;
        double r1408908 = r1408900 * r1408907;
        double r1408909 = sqrt(r1408908);
        double r1408910 = r1408898 * r1408909;
        return r1408910;
}
↓
double f(double re, double im) {
        double r1408911 = 0.5;
        double r1408912 = 2.0;
        double r1408913 = re;
        double r1408914 = r1408913 * r1408913;
        double r1408915 = /*Error: no posit support in C */;
        double r1408916 = im;
        double r1408917 = /*Error: no posit support in C */;
        double r1408918 = /*Error: no posit support in C */;
        double r1408919 = sqrt(r1408918);
        double r1408920 = r1408919 - r1408913;
        double r1408921 = r1408912 * r1408920;
        double r1408922 = sqrt(r1408921);
        double r1408923 = r1408911 * r1408922;
        return r1408923;
}

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. Using strategy rm

  3. Applied introduce-quire2.1

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

  4. Applied insert-quire-fdp-add2.1

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

  5. Final simplification2.1

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

Reproduce

herbie shell --seed 2019163 +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)))))