math.cube on complex, real part

Average Error: 7.7 → 0.2

Time: 2.3s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r272030 = x_re;
        double r272031 = r272030 * r272030;
        double r272032 = x_im;
        double r272033 = r272032 * r272032;
        double r272034 = r272031 - r272033;
        double r272035 = r272034 * r272030;
        double r272036 = r272030 * r272032;
        double r272037 = r272032 * r272030;
        double r272038 = r272036 + r272037;
        double r272039 = r272038 * r272032;
        double r272040 = r272035 - r272039;
        return r272040;
}

↓

double f(double x_re, double x_im) {
        double r272041 = x_re;
        double r272042 = 3.0;
        double r272043 = pow(r272041, r272042);
        double r272044 = x_im;
        double r272045 = r272041 * r272044;
        double r272046 = r272045 * r272044;
        double r272047 = r272042 * r272046;
        double r272048 = r272043 - r272047;
        return r272048;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.7
Target	0.3
Herbie	0.2

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

1. Initial program 7.7

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

2. Simplified 0.2

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

4. Applied associate-*r* 0.2

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

6. Applied associate-*r* 0.2

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

8. Applied associate-*l* 0.2

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

9. Simplified 0.2

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

11. Applied associate-*l* 0.2

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

12. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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