math.cube on complex, imaginary part

Average Error: 7.5 → 0.3

Time: 17.1s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r11611004 = x_re;
        double r11611005 = r11611004 * r11611004;
        double r11611006 = x_im;
        double r11611007 = r11611006 * r11611006;
        double r11611008 = r11611005 - r11611007;
        double r11611009 = r11611008 * r11611006;
        double r11611010 = r11611004 * r11611006;
        double r11611011 = r11611006 * r11611004;
        double r11611012 = r11611010 + r11611011;
        double r11611013 = r11611012 * r11611004;
        double r11611014 = r11611009 + r11611013;
        return r11611014;
}

↓

double f(double x_re, double x_im) {
        double r11611015 = x_re;
        double r11611016 = x_im;
        double r11611017 = r11611015 + r11611016;
        double r11611018 = r11611015 - r11611016;
        double r11611019 = r11611018 * r11611016;
        double r11611020 = r11611017 * r11611019;
        double r11611021 = r11611016 * r11611015;
        double r11611022 = r11611021 + r11611021;
        double r11611023 = r11611015 * r11611022;
        double r11611024 = r11611020 + r11611023;
        return r11611024;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
Target	0.3
Herbie	0.3

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

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* 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))