Average Error: 2.1 → 2.1
Time: 25.0s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} + re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\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 r326411 = 0.5;
        double r326412 = /* ERROR: no posit support in C */;
        double r326413 = 2.0;
        double r326414 = /* ERROR: no posit support in C */;
        double r326415 = re;
        double r326416 = r326415 * r326415;
        double r326417 = im;
        double r326418 = r326417 * r326417;
        double r326419 = r326416 + r326418;
        double r326420 = sqrt(r326419);
        double r326421 = r326420 + r326415;
        double r326422 = r326414 * r326421;
        double r326423 = sqrt(r326422);
        double r326424 = r326412 * r326423;
        return r326424;
}
↓
double f(double re, double im) {
        double r326425 = 0.5;
        double r326426 = 2.0;
        double r326427 = re;
        double r326428 = r326427 * r326427;
        double r326429 = /*Error: no posit support in C */;
        double r326430 = im;
        double r326431 = /*Error: no posit support in C */;
        double r326432 = /*Error: no posit support in C */;
        double r326433 = sqrt(r326432);
        double r326434 = r326433 + r326427;
        double r326435 = r326426 * r326434;
        double r326436 = sqrt(r326435);
        double r326437 = r326425 * r326436;
        return r326437;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

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

  3. Applied introduce-quire2.1

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

  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

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

Reproduce

herbie shell --seed 2019163 +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)))))