math.cube on complex, imaginary part

Average Error: 7.1 → 0.2

Time: 22.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r215942 = x_re;
        double r215943 = r215942 * r215942;
        double r215944 = x_im;
        double r215945 = r215944 * r215944;
        double r215946 = r215943 - r215945;
        double r215947 = r215946 * r215944;
        double r215948 = r215942 * r215944;
        double r215949 = r215944 * r215942;
        double r215950 = r215948 + r215949;
        double r215951 = r215950 * r215942;
        double r215952 = r215947 + r215951;
        return r215952;
}

↓

double f(double x_re, double x_im) {
        double r215953 = x_re;
        double r215954 = x_im;
        double r215955 = r215953 + r215954;
        double r215956 = r215955 * r215954;
        double r215957 = r215953 - r215954;
        double r215958 = r215956 * r215957;
        double r215959 = r215953 * r215954;
        double r215960 = r215954 * r215953;
        double r215961 = r215959 + r215960;
        double r215962 = r215961 * r215953;
        double r215963 = r215958 + r215962;
        return r215963;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
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 7.1

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

3. Applied difference-of-squares 7.1

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

4. Applied associate-*l* 0.2

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

6. Applied *-un-lft-identity 0.2

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

7. Applied associate-*l* 0.2

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

8. Simplified 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))