Average Error: 2.0 → 2.0
Time: 16.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
double f(double re, double im) {
        double r1104072 = 0.5;
        double r1104073 = 2.0;
        double r1104074 = re;
        double r1104075 = r1104074 * r1104074;
        double r1104076 = im;
        double r1104077 = r1104076 * r1104076;
        double r1104078 = r1104075 + r1104077;
        double r1104079 = sqrt(r1104078);
        double r1104080 = r1104079 - r1104074;
        double r1104081 = r1104073 * r1104080;
        double r1104082 = sqrt(r1104081);
        double r1104083 = r1104072 * r1104082;
        return r1104083;
}
↓
double f(double re, double im) {
        double r1104084 = 0.5;
        double r1104085 = 2.0;
        double r1104086 = re;
        double r1104087 = r1104086 * r1104086;
        double r1104088 = im;
        double r1104089 = r1104088 * r1104088;
        double r1104090 = r1104087 + r1104089;
        double r1104091 = sqrt(r1104090);
        double r1104092 = r1104091 - r1104086;
        double r1104093 = r1104085 * r1104092;
        double r1104094 = sqrt(r1104093);
        double r1104095 = r1104084 * r1104094;
        return r1104095;
}


0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}

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 2019101 +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)))))