Average Error: 2.0 → 2.0
Time: 5.4s
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 r11279 = 0.5;
        double r11280 = /* ERROR: no posit support in C */;
        double r11281 = 2.0;
        double r11282 = /* ERROR: no posit support in C */;
        double r11283 = re;
        double r11284 = r11283 * r11283;
        double r11285 = im;
        double r11286 = r11285 * r11285;
        double r11287 = r11284 + r11286;
        double r11288 = sqrt(r11287);
        double r11289 = r11288 + r11283;
        double r11290 = r11282 * r11289;
        double r11291 = sqrt(r11290);
        double r11292 = r11280 * r11291;
        return r11292;
}
↓
double f(double re, double im) {
        double r11293 = 0.5;
        double r11294 = 2.0;
        double r11295 = re;
        double r11296 = r11295 * r11295;
        double r11297 = im;
        double r11298 = r11297 * r11297;
        double r11299 = r11296 + r11298;
        double r11300 = sqrt(r11299);
        double r11301 = r11300 + r11295;
        double r11302 = r11294 * r11301;
        double r11303 = sqrt(r11302);
        double r11304 = r11293 * r11303;
        return r11304;
}

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 2019130 +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)))))