Average Error: 2.1 → 2.1
Time: 11.2s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} + re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\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 r154339 = 0.5;
        double r154340 = /* ERROR: no posit support in C */;
        double r154341 = 2.0;
        double r154342 = /* ERROR: no posit support in C */;
        double r154343 = re;
        double r154344 = r154343 * r154343;
        double r154345 = im;
        double r154346 = r154345 * r154345;
        double r154347 = r154344 + r154346;
        double r154348 = sqrt(r154347);
        double r154349 = r154348 + r154343;
        double r154350 = r154342 * r154349;
        double r154351 = sqrt(r154350);
        double r154352 = r154340 * r154351;
        return r154352;
}
↓
double f(double re, double im) {
        double r154353 = 0.5;
        double r154354 = 2.0;
        double r154355 = re;
        double r154356 = r154355 * r154355;
        double r154357 = /*Error: no posit support in C */;
        double r154358 = im;
        double r154359 = /*Error: no posit support in C */;
        double r154360 = /*Error: no posit support in C */;
        double r154361 = sqrt(r154360);
        double r154362 = r154361 + r154355;
        double r154363 = r154354 * r154362;
        double r154364 = sqrt(r154363);
        double r154365 = r154353 * r154364;
        return r154365;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied introduce-quire2.1

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

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

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

  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 2019154 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (+.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))