math.cube on complex, real part

Average Error: 7.3 → 0.2

Time: 23.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r107044 = x_re;
        double r107045 = r107044 * r107044;
        double r107046 = x_im;
        double r107047 = r107046 * r107046;
        double r107048 = r107045 - r107047;
        double r107049 = r107048 * r107044;
        double r107050 = r107044 * r107046;
        double r107051 = r107046 * r107044;
        double r107052 = r107050 + r107051;
        double r107053 = r107052 * r107046;
        double r107054 = r107049 - r107053;
        return r107054;
}

↓

double f(double x_re, double x_im) {
        double r107055 = x_re;
        double r107056 = 3.0;
        double r107057 = pow(r107055, r107056);
        double r107058 = x_im;
        double r107059 = r107058 * r107056;
        double r107060 = r107059 * r107055;
        double r107061 = r107060 * r107058;
        double r107062 = r107057 - r107061;
        return r107062;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.3
Target	0.2
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 7.3

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

2. Simplified 0.2

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

4. Applied associate-*r* 0.2

   \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im}\]
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. Simplified 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019323 
(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)))