Average Error: 2.1 → 2.2
Time: 20.5s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} \cdot 2.0 + \left(-re\right) \cdot 2.0}\]
\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)
↓
0.5 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} \cdot 2.0 + \left(-re\right) \cdot 2.0}
double f(double re, double im) {
        double r850158 = 0.5;
        double r850159 = /* ERROR: no posit support in C */;
        double r850160 = 2.0;
        double r850161 = /* ERROR: no posit support in C */;
        double r850162 = re;
        double r850163 = r850162 * r850162;
        double r850164 = im;
        double r850165 = r850164 * r850164;
        double r850166 = r850163 + r850165;
        double r850167 = sqrt(r850166);
        double r850168 = r850167 - r850162;
        double r850169 = r850161 * r850168;
        double r850170 = sqrt(r850169);
        double r850171 = r850159 * r850170;
        return r850171;
}
↓
double f(double re, double im) {
        double r850172 = 0.5;
        double r850173 = re;
        double r850174 = r850173 * r850173;
        double r850175 = im;
        double r850176 = r850175 * r850175;
        double r850177 = r850174 + r850176;
        double r850178 = sqrt(r850177);
        double r850179 = 2.0;
        double r850180 = r850178 * r850179;
        double r850181 = -r850173;
        double r850182 = r850181 * r850179;
        double r850183 = r850180 + r850182;
        double r850184 = sqrt(r850183);
        double r850185 = r850172 * r850184;
        return r850185;
}

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 sub-neg2.1

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

  4. Applied distribute-rgt-in2.2

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

  5. Final simplification2.2

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

Reproduce

herbie shell --seed 2019104 +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)))))