Average Error: 2.1 → 2.1
Time: 35.1s
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 r1327239 = 0.5;
        double r1327240 = /* ERROR: no posit support in C */;
        double r1327241 = 2.0;
        double r1327242 = /* ERROR: no posit support in C */;
        double r1327243 = re;
        double r1327244 = r1327243 * r1327243;
        double r1327245 = im;
        double r1327246 = r1327245 * r1327245;
        double r1327247 = r1327244 + r1327246;
        double r1327248 = sqrt(r1327247);
        double r1327249 = r1327248 - r1327243;
        double r1327250 = r1327242 * r1327249;
        double r1327251 = sqrt(r1327250);
        double r1327252 = r1327240 * r1327251;
        return r1327252;
}
↓
double f(double re, double im) {
        double r1327253 = 0.5;
        double r1327254 = 2.0;
        double r1327255 = re;
        double r1327256 = r1327255 * r1327255;
        double r1327257 = /*Error: no posit support in C */;
        double r1327258 = im;
        double r1327259 = /*Error: no posit support in C */;
        double r1327260 = /*Error: no posit support in C */;
        double r1327261 = sqrt(r1327260);
        double r1327262 = r1327261 - r1327255;
        double r1327263 = r1327254 * r1327262;
        double r1327264 = sqrt(r1327263);
        double r1327265 = r1327253 * r1327264;
        return r1327265;
}

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 2019163 
(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)))))