Average Error: 2.0 → 2.0
Time: 41.0s
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 r938366 = 0.5;
        double r938367 = /* ERROR: no posit support in C */;
        double r938368 = 2.0;
        double r938369 = /* ERROR: no posit support in C */;
        double r938370 = re;
        double r938371 = r938370 * r938370;
        double r938372 = im;
        double r938373 = r938372 * r938372;
        double r938374 = r938371 + r938373;
        double r938375 = sqrt(r938374);
        double r938376 = r938375 + r938370;
        double r938377 = r938369 * r938376;
        double r938378 = sqrt(r938377);
        double r938379 = r938367 * r938378;
        return r938379;
}
↓
double f(double re, double im) {
        double r938380 = 0.5;
        double r938381 = 2.0;
        double r938382 = re;
        double r938383 = r938382 * r938382;
        double r938384 = /*Error: no posit support in C */;
        double r938385 = im;
        double r938386 = /*Error: no posit support in C */;
        double r938387 = /*Error: no posit support in C */;
        double r938388 = sqrt(r938387);
        double r938389 = r938388 + r938382;
        double r938390 = r938381 * r938389;
        double r938391 = sqrt(r938390);
        double r938392 = r938380 * r938391;
        return r938392;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.0

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

  3. Applied introduce-quire2.0

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

  4. Applied insert-quire-fdp-add2.0

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

  5. Final simplification2.0

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

Reproduce

herbie shell --seed 2019158 +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)))))