Average Error: 2.0 → 2.0
Time: 5.5s
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 r14140 = 0.5;
        double r14141 = /* ERROR: no posit support in C */;
        double r14142 = 2.0;
        double r14143 = /* ERROR: no posit support in C */;
        double r14144 = re;
        double r14145 = r14144 * r14144;
        double r14146 = im;
        double r14147 = r14146 * r14146;
        double r14148 = r14145 + r14147;
        double r14149 = sqrt(r14148);
        double r14150 = r14149 + r14144;
        double r14151 = r14143 * r14150;
        double r14152 = sqrt(r14151);
        double r14153 = r14141 * r14152;
        return r14153;
}
↓
double f(double re, double im) {
        double r14154 = 0.5;
        double r14155 = 2.0;
        double r14156 = re;
        double r14157 = r14156 * r14156;
        double r14158 = im;
        double r14159 = r14158 * r14158;
        double r14160 = r14157 + r14159;
        double r14161 = sqrt(r14160);
        double r14162 = r14161 + r14156;
        double r14163 = r14155 * r14162;
        double r14164 = sqrt(r14163);
        double r14165 = r14154 * r14164;
        return r14165;
}

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)))))