Average Error: 2.2 → 2.1
Time: 40.1s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}\right)}{re}\right)\right)}\right)\]
\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)
↓
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}\right)}{re}\right)\right)}\right)
double f(double re, double im) {
        double r1020053 = 0.5;
        double r1020054 = /* ERROR: no posit support in C */;
        double r1020055 = 2.0;
        double r1020056 = /* ERROR: no posit support in C */;
        double r1020057 = re;
        double r1020058 = r1020057 * r1020057;
        double r1020059 = im;
        double r1020060 = r1020059 * r1020059;
        double r1020061 = r1020058 + r1020060;
        double r1020062 = sqrt(r1020061);
        double r1020063 = r1020062 + r1020057;
        double r1020064 = r1020056 * r1020063;
        double r1020065 = sqrt(r1020064);
        double r1020066 = r1020054 * r1020065;
        return r1020066;
}
↓
double f(double re, double im) {
        double r1020067 = 0.5;
        double r1020068 = /* ERROR: no posit support in C */;
        double r1020069 = 2.0;
        double r1020070 = /* ERROR: no posit support in C */;
        double r1020071 = re;
        double r1020072 = r1020071 * r1020071;
        double r1020073 = /*Error: no posit support in C */;
        double r1020074 = im;
        double r1020075 = /*Error: no posit support in C */;
        double r1020076 = /*Error: no posit support in C */;
        double r1020077 = sqrt(r1020076);
        double r1020078 = r1020077 + r1020071;
        double r1020079 = r1020070 * r1020078;
        double r1020080 = sqrt(r1020079);
        double r1020081 = r1020068 * r1020080;
        return r1020081;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.2

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied introduce-quire2.2

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

  4. Applied insert-quire-fdp-add2.1

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

  5. Final simplification2.1

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

Reproduce

herbie shell --seed 2019168 +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)))))