Average Error: 2.0 → 2.0
Time: 17.1s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]
\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{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}
double f(double re, double im) {
        double r556048 = 0.5;
        double r556049 = /* ERROR: no posit support in C */;
        double r556050 = 2.0;
        double r556051 = /* ERROR: no posit support in C */;
        double r556052 = re;
        double r556053 = r556052 * r556052;
        double r556054 = im;
        double r556055 = r556054 * r556054;
        double r556056 = r556053 + r556055;
        double r556057 = sqrt(r556056);
        double r556058 = r556057 - r556052;
        double r556059 = r556051 * r556058;
        double r556060 = sqrt(r556059);
        double r556061 = r556049 * r556060;
        return r556061;
}
↓
double f(double re, double im) {
        double r556062 = 0.5;
        double r556063 = 2.0;
        double r556064 = re;
        double r556065 = r556064 * r556064;
        double r556066 = /*Error: no posit support in C */;
        double r556067 = im;
        double r556068 = /*Error: no posit support in C */;
        double r556069 = /*Error: no posit support in C */;
        double r556070 = sqrt(r556069);
        double r556071 = r556070 - r556064;
        double r556072 = r556063 * r556071;
        double r556073 = sqrt(r556072);
        double r556074 = r556062 * r556073;
        return r556074;
}

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. Using strategy rm

  3. Applied introduce-quire2.0

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

  4. Applied insert-quire-fdp-add2.0

    \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}}\right) - re\right)\right)}\right)\]

  5. Final simplification2.0

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]

Reproduce

herbie shell --seed 2019154 +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)))))