Average Error: 2.0 → 2.0
Time: 35.3s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}\right) - re\right)\right)}\right)\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\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(\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 r1528484 = 0.5;
        double r1528485 = /* ERROR: no posit support in C */;
        double r1528486 = 2.0;
        double r1528487 = /* ERROR: no posit support in C */;
        double r1528488 = re;
        double r1528489 = r1528488 * r1528488;
        double r1528490 = im;
        double r1528491 = r1528490 * r1528490;
        double r1528492 = r1528489 + r1528491;
        double r1528493 = sqrt(r1528492);
        double r1528494 = r1528493 - r1528488;
        double r1528495 = r1528487 * r1528494;
        double r1528496 = sqrt(r1528495);
        double r1528497 = r1528485 * r1528496;
        return r1528497;
}
↓
double f(double re, double im) {
        double r1528498 = 0.5;
        double r1528499 = /* ERROR: no posit support in C */;
        double r1528500 = 2.0;
        double r1528501 = /* ERROR: no posit support in C */;
        double r1528502 = re;
        double r1528503 = r1528502 * r1528502;
        double r1528504 = /*Error: no posit support in C */;
        double r1528505 = im;
        double r1528506 = /*Error: no posit support in C */;
        double r1528507 = /*Error: no posit support in C */;
        double r1528508 = sqrt(r1528507);
        double r1528509 = r1528508 - r1528502;
        double r1528510 = r1528501 * r1528509;
        double r1528511 = sqrt(r1528510);
        double r1528512 = r1528499 * r1528511;
        return r1528512;
}

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

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, 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)))))