Average Error: 2.0 → 2.0
Time: 5.9s
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 r11296 = 0.5;
        double r11297 = /* ERROR: no posit support in C */;
        double r11298 = 2.0;
        double r11299 = /* ERROR: no posit support in C */;
        double r11300 = re;
        double r11301 = r11300 * r11300;
        double r11302 = im;
        double r11303 = r11302 * r11302;
        double r11304 = r11301 + r11303;
        double r11305 = sqrt(r11304);
        double r11306 = r11305 + r11300;
        double r11307 = r11299 * r11306;
        double r11308 = sqrt(r11307);
        double r11309 = r11297 * r11308;
        return r11309;
}
↓
double f(double re, double im) {
        double r11310 = 0.5;
        double r11311 = 2.0;
        double r11312 = re;
        double r11313 = r11312 * r11312;
        double r11314 = im;
        double r11315 = r11314 * r11314;
        double r11316 = r11313 + r11315;
        double r11317 = sqrt(r11316);
        double r11318 = r11317 + r11312;
        double r11319 = r11311 * r11318;
        double r11320 = sqrt(r11319);
        double r11321 = r11310 * r11320;
        return r11321;
}

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. Final simplification2.0

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

Reproduce

herbie shell --seed 2019134 
(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)))))