Average Error: 2.1 → 2.1
Time: 10.5s
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 r152432 = 0.5;
        double r152433 = /* ERROR: no posit support in C */;
        double r152434 = 2.0;
        double r152435 = /* ERROR: no posit support in C */;
        double r152436 = re;
        double r152437 = r152436 * r152436;
        double r152438 = im;
        double r152439 = r152438 * r152438;
        double r152440 = r152437 + r152439;
        double r152441 = sqrt(r152440);
        double r152442 = r152441 + r152436;
        double r152443 = r152435 * r152442;
        double r152444 = sqrt(r152443);
        double r152445 = r152433 * r152444;
        return r152445;
}
↓
double f(double re, double im) {
        double r152446 = 0.5;
        double r152447 = 2.0;
        double r152448 = re;
        double r152449 = r152448 * r152448;
        double r152450 = /*Error: no posit support in C */;
        double r152451 = im;
        double r152452 = /*Error: no posit support in C */;
        double r152453 = /*Error: no posit support in C */;
        double r152454 = sqrt(r152453);
        double r152455 = r152454 + r152448;
        double r152456 = r152447 * r152455;
        double r152457 = sqrt(r152456);
        double r152458 = r152446 * r152457;
        return r152458;
}

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 2019156 +o rules:numerics
(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)))))