math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 22.0s

Precision: 64

Math

\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]

↓

\[\left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)\]

C

double f(double x_re, double x_im) {
        double r622093 = x_re;
        double r622094 = r622093 * r622093;
        double r622095 = x_im;
        double r622096 = r622095 * r622095;
        double r622097 = r622094 - r622096;
        double r622098 = r622097 * r622093;
        double r622099 = r622093 * r622095;
        double r622100 = r622095 * r622093;
        double r622101 = r622099 + r622100;
        double r622102 = r622101 * r622095;
        double r622103 = r622098 - r622102;
        return r622103;
}

↓

double f(double x_re, double x_im) {
        double r622104 = x_re;
        double r622105 = x_im;
        double r622106 = r622104 + r622105;
        double r622107 = r622104 * r622106;
        double r622108 = r622104 - r622105;
        double r622109 = r622107 * r622108;
        double r622110 = /*Error: no posit support in C */;
        double r622111 = r622104 + r622104;
        double r622112 = r622105 * r622111;
        double r622113 = /*Error: no posit support in C */;
        double r622114 = /*Error: no posit support in C */;
        return r622114;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

1. Initial program 0.4

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

3. Applied introduce-quire 0.4

   \[\leadsto \color{blue}{\left(\left(\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right)\right)\right)} - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]

4. Applied insert-quire-fdp-sub 0.3

   \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right)\right), \left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right), x.im\right)\right)}\]

5. Simplified 0.3

   \[\leadsto \color{blue}{\left(\mathsf{qms}\left(\left(\left(x.re \cdot \left(\left(\frac{x.re}{x.im}\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(\frac{\left(x.re \cdot x.im\right)}{\left(x.re \cdot x.im\right)}\right), x.im\right)\right)}\]
6. Using strategy rm

7. Applied distribute-rgt-out 0.3

   \[\leadsto \left(\mathsf{qms}\left(\left(\left(x.re \cdot \left(\left(\frac{x.re}{x.im}\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \color{blue}{\left(x.im \cdot \left(\frac{x.re}{x.re}\right)\right)}, x.im\right)\right)\]
8. Using strategy rm

9. Applied associate-*r* 0.3

   \[\leadsto \left(\mathsf{qms}\left(\left(\color{blue}{\left(\left(x.re \cdot \left(\frac{x.re}{x.im}\right)\right) \cdot \left(x.re - x.im\right)\right)}\right), \left(x.im \cdot \left(\frac{x.re}{x.re}\right)\right), x.im\right)\right)\]

10. Final simplification 0.3

    \[\leadsto \left(\mathsf{qms}\left(\left(\left(\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.im\right)\right)\]

Reproduce

herbie shell --seed 2019155 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))