Average Error: 2.1 → 2.1
Time: 6.6s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
\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{re \cdot re + im \cdot im} + re\right)}
double f(double re, double im) {
        double r11323 = 0.5;
        double r11324 = /* ERROR: no posit support in C */;
        double r11325 = 2.0;
        double r11326 = /* ERROR: no posit support in C */;
        double r11327 = re;
        double r11328 = r11327 * r11327;
        double r11329 = im;
        double r11330 = r11329 * r11329;
        double r11331 = r11328 + r11330;
        double r11332 = sqrt(r11331);
        double r11333 = r11332 + r11327;
        double r11334 = r11326 * r11333;
        double r11335 = sqrt(r11334);
        double r11336 = r11324 * r11335;
        return r11336;
}
↓
double f(double re, double im) {
        double r11337 = 0.5;
        double r11338 = 2.0;
        double r11339 = re;
        double r11340 = r11339 * r11339;
        double r11341 = im;
        double r11342 = r11341 * r11341;
        double r11343 = r11340 + r11342;
        double r11344 = sqrt(r11343);
        double r11345 = r11344 + r11339;
        double r11346 = r11338 * r11345;
        double r11347 = sqrt(r11346);
        double r11348 = r11337 * r11347;
        return r11348;
}

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. Final simplification2.1

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

Reproduce

herbie shell --seed 2019135 +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)))))