math.cube on complex, imaginary part

Average Error: 7.5 → 0.2

Time: 14.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r337014 = x_re;
        double r337015 = r337014 * r337014;
        double r337016 = x_im;
        double r337017 = r337016 * r337016;
        double r337018 = r337015 - r337017;
        double r337019 = r337018 * r337016;
        double r337020 = r337014 * r337016;
        double r337021 = r337016 * r337014;
        double r337022 = r337020 + r337021;
        double r337023 = r337022 * r337014;
        double r337024 = r337019 + r337023;
        return r337024;
}

↓

double f(double x_re, double x_im) {
        double r337025 = x_re;
        double r337026 = x_im;
        double r337027 = r337025 * r337026;
        double r337028 = r337026 * r337025;
        double r337029 = r337027 + r337028;
        double r337030 = r337025 * r337029;
        double r337031 = r337025 + r337026;
        double r337032 = r337025 - r337026;
        double r337033 = r337032 * r337026;
        double r337034 = r337031 * r337033;
        double r337035 = r337030 + r337034;
        return r337035;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
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 7.5

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

3. Applied difference-of-squares 7.5

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

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

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019351 
(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)))