math.cube on complex, real part

Average Error: 7.2 → 0.3

Time: 26.9s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r9153307 = x_re;
        double r9153308 = r9153307 * r9153307;
        double r9153309 = x_im;
        double r9153310 = r9153309 * r9153309;
        double r9153311 = r9153308 - r9153310;
        double r9153312 = r9153311 * r9153307;
        double r9153313 = r9153307 * r9153309;
        double r9153314 = r9153309 * r9153307;
        double r9153315 = r9153313 + r9153314;
        double r9153316 = r9153315 * r9153309;
        double r9153317 = r9153312 - r9153316;
        return r9153317;
}

↓

double f(double x_re, double x_im) {
        double r9153318 = x_re;
        double r9153319 = x_im;
        double r9153320 = r9153318 - r9153319;
        double r9153321 = r9153320 * r9153318;
        double r9153322 = r9153319 + r9153318;
        double r9153323 = r9153321 * r9153322;
        double r9153324 = r9153318 * r9153319;
        double r9153325 = r9153324 + r9153324;
        double r9153326 = r9153325 * r9153319;
        double r9153327 = r9153323 - r9153326;
        return r9153327;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 7.2

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

3. Applied difference-of-squares 7.2

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

4. Applied associate-*l* 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019169 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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