Average Error: 2.1 → 2.1
Time: 34.6s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]
\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 r1452311 = 0.5;
        double r1452312 = /* ERROR: no posit support in C */;
        double r1452313 = 2.0;
        double r1452314 = /* ERROR: no posit support in C */;
        double r1452315 = re;
        double r1452316 = r1452315 * r1452315;
        double r1452317 = im;
        double r1452318 = r1452317 * r1452317;
        double r1452319 = r1452316 + r1452318;
        double r1452320 = sqrt(r1452319);
        double r1452321 = r1452320 - r1452315;
        double r1452322 = r1452314 * r1452321;
        double r1452323 = sqrt(r1452322);
        double r1452324 = r1452312 * r1452323;
        return r1452324;
}
↓
double f(double re, double im) {
        double r1452325 = 0.5;
        double r1452326 = 2.0;
        double r1452327 = re;
        double r1452328 = r1452327 * r1452327;
        double r1452329 = /*Error: no posit support in C */;
        double r1452330 = im;
        double r1452331 = /*Error: no posit support in C */;
        double r1452332 = /*Error: no posit support in C */;
        double r1452333 = sqrt(r1452332);
        double r1452334 = r1452333 - r1452327;
        double r1452335 = r1452326 * r1452334;
        double r1452336 = sqrt(r1452335);
        double r1452337 = r1452325 * r1452336;
        return r1452337;
}

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 2019162 
(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)))))