math.cube on complex, real part

Average Error: 6.9 → 0.2

Time: 30.9s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(-x.im, x.re \cdot x.im + x.re \cdot x.im, x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right) + \mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.re, \left(-x.im\right) \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

C

double f(double x_re, double x_im) {
        double r8740026 = x_re;
        double r8740027 = r8740026 * r8740026;
        double r8740028 = x_im;
        double r8740029 = r8740028 * r8740028;
        double r8740030 = r8740027 - r8740029;
        double r8740031 = r8740030 * r8740026;
        double r8740032 = r8740026 * r8740028;
        double r8740033 = r8740028 * r8740026;
        double r8740034 = r8740032 + r8740033;
        double r8740035 = r8740034 * r8740028;
        double r8740036 = r8740031 - r8740035;
        return r8740036;
}

↓

double f(double x_re, double x_im) {
        double r8740037 = x_im;
        double r8740038 = -r8740037;
        double r8740039 = x_re;
        double r8740040 = r8740039 * r8740037;
        double r8740041 = r8740040 + r8740040;
        double r8740042 = r8740037 * r8740041;
        double r8740043 = fma(r8740038, r8740041, r8740042);
        double r8740044 = r8740037 + r8740039;
        double r8740045 = r8740039 - r8740037;
        double r8740046 = r8740045 * r8740039;
        double r8740047 = r8740038 * r8740041;
        double r8740048 = fma(r8740044, r8740046, r8740047);
        double r8740049 = r8740043 + r8740048;
        return r8740049;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
Target	0.3
Herbie	0.2

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

Derivation

1. Initial program 6.9

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

3. Applied difference-of-squares 6.9

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

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. Using strategy rm

6. Applied prod-diff 0.2

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

7. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(-x.im, x.re \cdot x.im + x.re \cdot x.im, x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right) + \mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.re, \left(-x.im\right) \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(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 x.im))))

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