Average Error: 2.1 → 2.1
Time: 10.7s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\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 r228040 = 0.5;
        double r228041 = /* ERROR: no posit support in C */;
        double r228042 = 2.0;
        double r228043 = /* ERROR: no posit support in C */;
        double r228044 = re;
        double r228045 = r228044 * r228044;
        double r228046 = im;
        double r228047 = r228046 * r228046;
        double r228048 = r228045 + r228047;
        double r228049 = sqrt(r228048);
        double r228050 = r228049 - r228044;
        double r228051 = r228043 * r228050;
        double r228052 = sqrt(r228051);
        double r228053 = r228041 * r228052;
        return r228053;
}
↓
double f(double re, double im) {
        double r228054 = 0.5;
        double r228055 = 2.0;
        double r228056 = re;
        double r228057 = r228056 * r228056;
        double r228058 = im;
        double r228059 = r228058 * r228058;
        double r228060 = r228057 + r228059;
        double r228061 = sqrt(r228060);
        double r228062 = r228061 - r228056;
        double r228063 = r228055 * r228062;
        double r228064 = sqrt(r228063);
        double r228065 = r228054 * r228064;
        return r228065;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]

  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 2019132 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (-.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))