math.cube on complex, imaginary part

Average Error: 6.9 → 0.2

Time: 20.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r6452958 = x_re;
        double r6452959 = r6452958 * r6452958;
        double r6452960 = x_im;
        double r6452961 = r6452960 * r6452960;
        double r6452962 = r6452959 - r6452961;
        double r6452963 = r6452962 * r6452960;
        double r6452964 = r6452958 * r6452960;
        double r6452965 = r6452960 * r6452958;
        double r6452966 = r6452964 + r6452965;
        double r6452967 = r6452966 * r6452958;
        double r6452968 = r6452963 + r6452967;
        return r6452968;
}

↓

double f(double x_re, double x_im) {
        double r6452969 = x_re;
        double r6452970 = 3.0;
        double r6452971 = r6452969 * r6452970;
        double r6452972 = x_im;
        double r6452973 = r6452971 * r6452972;
        double r6452974 = r6452973 * r6452969;
        double r6452975 = pow(r6452972, r6452970);
        double r6452976 = r6452974 - r6452975;
        return r6452976;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
Target	0.3
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 6.9

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

2. Taylor expanded around 0 6.9

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

3. Simplified 0.3

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

5. Applied pow1 0.3

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

6. Applied pow1 0.3

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

7. Applied pow-prod-up 0.3

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

8. Applied pow1 0.3

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

9. Applied pow-prod-up 0.2

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

10. Simplified 0.2

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

11. Taylor expanded around 0 6.9

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

12. Simplified 0.2

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

14. Applied associate-*r* 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 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)))