Average Error: 2.0 → 2.0
Time: 5.3s
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 r11335 = 0.5;
        double r11336 = /* ERROR: no posit support in C */;
        double r11337 = 2.0;
        double r11338 = /* ERROR: no posit support in C */;
        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 = r11336 * r11347;
        return r11348;
}
↓
double f(double re, double im) {
        double r11349 = 0.5;
        double r11350 = 2.0;
        double r11351 = re;
        double r11352 = r11351 * r11351;
        double r11353 = im;
        double r11354 = r11353 * r11353;
        double r11355 = r11352 + r11354;
        double r11356 = sqrt(r11355);
        double r11357 = r11356 + r11351;
        double r11358 = r11350 * r11357;
        double r11359 = sqrt(r11358);
        double r11360 = r11349 * r11359;
        return r11360;
}

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 2019128 +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)))))