Average Error: 2.0 → 2.0
Time: 9.5s
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 r228031 = 0.5;
        double r228032 = /* ERROR: no posit support in C */;
        double r228033 = 2.0;
        double r228034 = /* ERROR: no posit support in C */;
        double r228035 = re;
        double r228036 = r228035 * r228035;
        double r228037 = im;
        double r228038 = r228037 * r228037;
        double r228039 = r228036 + r228038;
        double r228040 = sqrt(r228039);
        double r228041 = r228040 - r228035;
        double r228042 = r228034 * r228041;
        double r228043 = sqrt(r228042);
        double r228044 = r228032 * r228043;
        return r228044;
}
↓
double f(double re, double im) {
        double r228045 = 0.5;
        double r228046 = 2.0;
        double r228047 = re;
        double r228048 = r228047 * r228047;
        double r228049 = im;
        double r228050 = r228049 * r228049;
        double r228051 = r228048 + r228050;
        double r228052 = sqrt(r228051);
        double r228053 = r228052 - r228047;
        double r228054 = r228046 * r228053;
        double r228055 = sqrt(r228054);
        double r228056 = r228045 * r228055;
        return r228056;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.0

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

  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 2019133 
(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)))))