math.cube on complex, imaginary part

Average Error: 7.0 → 0.2

Time: 18.5s

Precision: 64

Math

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

↓

\[x.re \cdot \left(\left(x.im \cdot 3\right) \cdot x.re\right) - {x.im}^{3}\]

C

double f(double x_re, double x_im) {
        double r8125164 = x_re;
        double r8125165 = r8125164 * r8125164;
        double r8125166 = x_im;
        double r8125167 = r8125166 * r8125166;
        double r8125168 = r8125165 - r8125167;
        double r8125169 = r8125168 * r8125166;
        double r8125170 = r8125164 * r8125166;
        double r8125171 = r8125166 * r8125164;
        double r8125172 = r8125170 + r8125171;
        double r8125173 = r8125172 * r8125164;
        double r8125174 = r8125169 + r8125173;
        return r8125174;
}

↓

double f(double x_re, double x_im) {
        double r8125175 = x_re;
        double r8125176 = x_im;
        double r8125177 = 3.0;
        double r8125178 = r8125176 * r8125177;
        double r8125179 = r8125178 * r8125175;
        double r8125180 = r8125175 * r8125179;
        double r8125181 = pow(r8125176, r8125177);
        double r8125182 = r8125180 - r8125181;
        return r8125182;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.2
Herbie	0.2

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

1. Initial program 7.0

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

2. Simplified 7.0

   \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)}\]

3. Taylor expanded around 0 6.9

   \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^{2}\right) - {x.im}^{3}}\]

4. Simplified 0.2

   \[\leadsto \color{blue}{\left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \left(x.im \cdot x.im\right)}\]
5. Using strategy rm

6. Applied pow1 0.2

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \left(x.im \cdot \color{blue}{{x.im}^{1}}\right)\]

7. Applied pow1 0.2

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \left(\color{blue}{{x.im}^{1}} \cdot {x.im}^{1}\right)\]

8. Applied pow-prod-up 0.2

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - x.im \cdot \color{blue}{{x.im}^{\left(1 + 1\right)}}\]

9. Applied pow1 0.2

   \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{1}} \cdot {x.im}^{\left(1 + 1\right)}\]

10. Applied pow-prod-up 0.2

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{\left(1 + \left(1 + 1\right)\right)}}\]

11. Simplified 0.2

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - {x.im}^{\color{blue}{3}}\]

12. Final simplification 0.2

    \[\leadsto x.re \cdot \left(\left(x.im \cdot 3\right) \cdot x.re\right) - {x.im}^{3}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))