math.cube on complex, real part

Average Error: 0.4 → 0.3

Time: 35.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r3163120 = x_re;
        double r3163121 = r3163120 * r3163120;
        double r3163122 = x_im;
        double r3163123 = r3163122 * r3163122;
        double r3163124 = r3163121 - r3163123;
        double r3163125 = r3163124 * r3163120;
        double r3163126 = r3163120 * r3163122;
        double r3163127 = r3163122 * r3163120;
        double r3163128 = r3163126 + r3163127;
        double r3163129 = r3163128 * r3163122;
        double r3163130 = r3163125 - r3163129;
        return r3163130;
}

↓

double f(double x_re, double x_im) {
        double r3163131 = x_re;
        double r3163132 = x_im;
        double r3163133 = r3163131 - r3163132;
        double r3163134 = r3163131 * r3163133;
        double r3163135 = r3163132 + r3163131;
        double r3163136 = r3163134 * r3163135;
        double r3163137 = /*Error: no posit support in C */;
        double r3163138 = r3163132 * r3163131;
        double r3163139 = r3163138 + r3163138;
        double r3163140 = /*Error: no posit support in C */;
        double r3163141 = /*Error: no posit support in C */;
        return r3163141;
}

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

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

7. Applied associate-*r* 0.3

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

9. Applied distribute-rgt-in 0.3

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

10. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(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)))