Average Error: 2.1 → 2.1
Time: 48.0s
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 r1395106 = 0.5;
        double r1395107 = /* ERROR: no posit support in C */;
        double r1395108 = 2.0;
        double r1395109 = /* ERROR: no posit support in C */;
        double r1395110 = re;
        double r1395111 = r1395110 * r1395110;
        double r1395112 = im;
        double r1395113 = r1395112 * r1395112;
        double r1395114 = r1395111 + r1395113;
        double r1395115 = sqrt(r1395114);
        double r1395116 = r1395115 - r1395110;
        double r1395117 = r1395109 * r1395116;
        double r1395118 = sqrt(r1395117);
        double r1395119 = r1395107 * r1395118;
        return r1395119;
}
↓
double f(double re, double im) {
        double r1395120 = 0.5;
        double r1395121 = 2.0;
        double r1395122 = re;
        double r1395123 = r1395122 * r1395122;
        double r1395124 = /*Error: no posit support in C */;
        double r1395125 = im;
        double r1395126 = /*Error: no posit support in C */;
        double r1395127 = /*Error: no posit support in C */;
        double r1395128 = sqrt(r1395127);
        double r1395129 = r1395128 - r1395122;
        double r1395130 = r1395121 * r1395129;
        double r1395131 = sqrt(r1395130);
        double r1395132 = r1395120 * r1395131;
        return r1395132;
}

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. Using strategy rm

  3. Applied introduce-quire2.1

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

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

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

  5. Final simplification2.1

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

Reproduce

herbie shell --seed 2019165 
(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)))))