math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 2.1 → 1.2

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le 1.57421875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot \left(im \cdot im\right)}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\\ \end{array}\]

C

double f(double re, double im) {
        double r1074037 = 0.5;
        double r1074038 = /* ERROR: no posit support in C */;
        double r1074039 = 2.0;
        double r1074040 = /* ERROR: no posit support in C */;
        double r1074041 = re;
        double r1074042 = r1074041 * r1074041;
        double r1074043 = im;
        double r1074044 = r1074043 * r1074043;
        double r1074045 = r1074042 + r1074044;
        double r1074046 = sqrt(r1074045);
        double r1074047 = r1074046 - r1074041;
        double r1074048 = r1074040 * r1074047;
        double r1074049 = sqrt(r1074048);
        double r1074050 = r1074038 * r1074049;
        return r1074050;
}

↓

double f(double re, double im) {
        double r1074051 = re;
        double r1074052 = 1.57421875;
        bool r1074053 = r1074051 <= r1074052;
        double r1074054 = 0.5;
        double r1074055 = 2.0;
        double r1074056 = r1074051 * r1074051;
        double r1074057 = im;
        double r1074058 = r1074057 * r1074057;
        double r1074059 = r1074056 + r1074058;
        double r1074060 = sqrt(r1074059);
        double r1074061 = r1074060 - r1074051;
        double r1074062 = r1074055 * r1074061;
        double r1074063 = sqrt(r1074062);
        double r1074064 = r1074054 * r1074063;
        double r1074065 = r1074058 + r1074056;
        double r1074066 = sqrt(r1074065);
        double r1074067 = r1074051 + r1074066;
        double r1074068 = r1074067 * r1074058;
        double r1074069 = r1074060 + r1074051;
        double r1074070 = r1074069 * r1074069;
        double r1074071 = r1074068 / r1074070;
        double r1074072 = r1074055 * r1074071;
        double r1074073 = sqrt(r1074072);
        double r1074074 = r1074054 * r1074073;
        double r1074075 = r1074053 ? r1074064 : r1074074;
        return r1074075;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 2 regimes

2. if re < 1.57421875

   1. Initial program 0.9

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

   if 1.57421875 < re

   1. Initial program 6.2

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

   3. Applied p16-flip-- 5.5

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

   5. Applied difference-of-squares 6.3

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

   7. Applied p16-flip-- 5.5

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

   8. Applied associate-*r/ 6.0

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

   9. Applied associate-/l/ 6.0

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

   10. Simplified 2.1

       \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(\left(\frac{re}{\left(\sqrt{\left(\frac{\left(im \cdot im\right)}{\left(re \cdot re\right)}\right)}\right)}\right) \cdot \left(im \cdot im\right)\right)}}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.57421875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot \left(im \cdot im\right)}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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)))))