Average Error: 2.1 → 2.1
Time: 7.0s
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 r15020 = 0.5;
        double r15021 = /* ERROR: no posit support in C */;
        double r15022 = 2.0;
        double r15023 = /* ERROR: no posit support in C */;
        double r15024 = re;
        double r15025 = r15024 * r15024;
        double r15026 = im;
        double r15027 = r15026 * r15026;
        double r15028 = r15025 + r15027;
        double r15029 = sqrt(r15028);
        double r15030 = r15029 + r15024;
        double r15031 = r15023 * r15030;
        double r15032 = sqrt(r15031);
        double r15033 = r15021 * r15032;
        return r15033;
}
↓
double f(double re, double im) {
        double r15034 = 0.5;
        double r15035 = 2.0;
        double r15036 = re;
        double r15037 = r15036 * r15036;
        double r15038 = im;
        double r15039 = r15038 * r15038;
        double r15040 = r15037 + r15039;
        double r15041 = sqrt(r15040);
        double r15042 = r15041 + r15036;
        double r15043 = r15035 * r15042;
        double r15044 = sqrt(r15043);
        double r15045 = r15034 * r15044;
        return r15045;
}

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 2019107 +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)))))