math.cube on complex, imaginary part

Average Error: 7.1 → 0.8

Time: 36.2s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(x.im + x.re, \left(\left(x.re - x.im\right) \cdot \left(\sqrt[3]{x.im} \cdot \sqrt[3]{x.im}\right)\right) \cdot \sqrt[3]{x.im}, x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

C

double f(double x_re, double x_im) {
        double r6378138 = x_re;
        double r6378139 = r6378138 * r6378138;
        double r6378140 = x_im;
        double r6378141 = r6378140 * r6378140;
        double r6378142 = r6378139 - r6378141;
        double r6378143 = r6378142 * r6378140;
        double r6378144 = r6378138 * r6378140;
        double r6378145 = r6378140 * r6378138;
        double r6378146 = r6378144 + r6378145;
        double r6378147 = r6378146 * r6378138;
        double r6378148 = r6378143 + r6378147;
        return r6378148;
}

↓

double f(double x_re, double x_im) {
        double r6378149 = x_im;
        double r6378150 = x_re;
        double r6378151 = r6378149 + r6378150;
        double r6378152 = r6378150 - r6378149;
        double r6378153 = cbrt(r6378149);
        double r6378154 = r6378153 * r6378153;
        double r6378155 = r6378152 * r6378154;
        double r6378156 = r6378155 * r6378153;
        double r6378157 = r6378150 * r6378149;
        double r6378158 = r6378157 + r6378157;
        double r6378159 = r6378150 * r6378158;
        double r6378160 = fma(r6378151, r6378156, r6378159);
        return r6378160;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.8

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

Derivation

1. Initial program 7.1

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

3. Applied difference-of-squares 7.1

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

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. Using strategy rm

6. Applied fma-def 0.2

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

8. Applied add-cube-cbrt 0.8

   \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x.im} \cdot \sqrt[3]{x.im}\right) \cdot \sqrt[3]{x.im}\right)}, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\]

9. Applied associate-*r* 0.8

   \[\leadsto \mathsf{fma}\left(x.re + x.im, \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(\sqrt[3]{x.im} \cdot \sqrt[3]{x.im}\right)\right) \cdot \sqrt[3]{x.im}}, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\]

10. Final simplification 0.8

    \[\leadsto \mathsf{fma}\left(x.im + x.re, \left(\left(x.re - x.im\right) \cdot \left(\sqrt[3]{x.im} \cdot \sqrt[3]{x.im}\right)\right) \cdot \sqrt[3]{x.im}, x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :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)))