math.cube on complex, real part

Average Error: 0.4 → 0.4

Time: 36.2s

Precision: 64

Math

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

↓

\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(-\left(\left(x.im \cdot x.re\right) \cdot x.im + \left(x.im \cdot x.re\right) \cdot x.im\right)\right)\]

C

double f(double x_re, double x_im) {
        double r4176025 = x_re;
        double r4176026 = r4176025 * r4176025;
        double r4176027 = x_im;
        double r4176028 = r4176027 * r4176027;
        double r4176029 = r4176026 - r4176028;
        double r4176030 = r4176029 * r4176025;
        double r4176031 = r4176025 * r4176027;
        double r4176032 = r4176027 * r4176025;
        double r4176033 = r4176031 + r4176032;
        double r4176034 = r4176033 * r4176027;
        double r4176035 = r4176030 - r4176034;
        return r4176035;
}

↓

double f(double x_re, double x_im) {
        double r4176036 = x_re;
        double r4176037 = x_im;
        double r4176038 = r4176036 + r4176037;
        double r4176039 = r4176036 - r4176037;
        double r4176040 = r4176039 * r4176036;
        double r4176041 = r4176038 * r4176040;
        double r4176042 = r4176037 * r4176036;
        double r4176043 = r4176042 * r4176037;
        double r4176044 = r4176043 + r4176043;
        double r4176045 = -r4176044;
        double r4176046 = r4176041 + r4176045;
        return r4176046;
}

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 difference-of-squares 0.4

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

4. Applied associate-*l* 0.4

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

6. Applied sub-neg 0.4

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

7. Simplified 0.4

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

8. Final simplification 0.4

   \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(-\left(\left(x.im \cdot x.re\right) \cdot x.im + \left(x.im \cdot x.re\right) \cdot x.im\right)\right)\]

Reproduce

herbie shell --seed 2019133 
(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)))