Average Error: 2.1 → 2.1
Time: 38.1s
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 r1348246 = 0.5;
        double r1348247 = /* ERROR: no posit support in C */;
        double r1348248 = 2.0;
        double r1348249 = /* ERROR: no posit support in C */;
        double r1348250 = re;
        double r1348251 = r1348250 * r1348250;
        double r1348252 = im;
        double r1348253 = r1348252 * r1348252;
        double r1348254 = r1348251 + r1348253;
        double r1348255 = sqrt(r1348254);
        double r1348256 = r1348255 - r1348250;
        double r1348257 = r1348249 * r1348256;
        double r1348258 = sqrt(r1348257);
        double r1348259 = r1348247 * r1348258;
        return r1348259;
}
↓
double f(double re, double im) {
        double r1348260 = 0.5;
        double r1348261 = 2.0;
        double r1348262 = re;
        double r1348263 = r1348262 * r1348262;
        double r1348264 = /*Error: no posit support in C */;
        double r1348265 = im;
        double r1348266 = /*Error: no posit support in C */;
        double r1348267 = /*Error: no posit support in C */;
        double r1348268 = sqrt(r1348267);
        double r1348269 = r1348268 - r1348262;
        double r1348270 = r1348261 * r1348269;
        double r1348271 = sqrt(r1348270);
        double r1348272 = r1348260 * r1348271;
        return r1348272;
}

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 
(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)))))