math.cube on complex, imaginary part

Average Error: 7.6 → 0.2

Time: 26.6s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r179288 = x_re;
        double r179289 = r179288 * r179288;
        double r179290 = x_im;
        double r179291 = r179290 * r179290;
        double r179292 = r179289 - r179291;
        double r179293 = r179292 * r179290;
        double r179294 = r179288 * r179290;
        double r179295 = r179290 * r179288;
        double r179296 = r179294 + r179295;
        double r179297 = r179296 * r179288;
        double r179298 = r179293 + r179297;
        return r179298;
}

↓

double f(double x_re, double x_im) {
        double r179299 = x_re;
        double r179300 = x_im;
        double r179301 = r179299 + r179300;
        double r179302 = r179299 - r179300;
        double r179303 = r179302 * r179300;
        double r179304 = r179301 * r179303;
        double r179305 = r179299 * r179300;
        double r179306 = r179300 * r179299;
        double r179307 = r179305 + r179306;
        double r179308 = r179307 * r179299;
        double r179309 = r179304 + r179308;
        return r179309;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.6
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 7.6

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

3. Applied difference-of-squares 7.6

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

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019347 
(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)))