Average Error: 2.1 → 2.0
Time: 13.8s
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 r154226 = 0.5;
        double r154227 = /* ERROR: no posit support in C */;
        double r154228 = 2.0;
        double r154229 = /* ERROR: no posit support in C */;
        double r154230 = re;
        double r154231 = r154230 * r154230;
        double r154232 = im;
        double r154233 = r154232 * r154232;
        double r154234 = r154231 + r154233;
        double r154235 = sqrt(r154234);
        double r154236 = r154235 + r154230;
        double r154237 = r154229 * r154236;
        double r154238 = sqrt(r154237);
        double r154239 = r154227 * r154238;
        return r154239;
}
↓
double f(double re, double im) {
        double r154240 = 0.5;
        double r154241 = 2.0;
        double r154242 = re;
        double r154243 = r154242 * r154242;
        double r154244 = /*Error: no posit support in C */;
        double r154245 = im;
        double r154246 = /*Error: no posit support in C */;
        double r154247 = /*Error: no posit support in C */;
        double r154248 = sqrt(r154247);
        double r154249 = r154248 + r154242;
        double r154250 = r154241 * r154249;
        double r154251 = sqrt(r154250);
        double r154252 = r154240 * r154251;
        return r154252;
}

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-add2.0

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

  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 2019155 
(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)))))