math.cube on complex, real part

Average Error: 6.6 → 0.6

Time: 21.7s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r7212162 = x_re;
        double r7212163 = r7212162 * r7212162;
        double r7212164 = x_im;
        double r7212165 = r7212164 * r7212164;
        double r7212166 = r7212163 - r7212165;
        double r7212167 = r7212166 * r7212162;
        double r7212168 = r7212162 * r7212164;
        double r7212169 = r7212164 * r7212162;
        double r7212170 = r7212168 + r7212169;
        double r7212171 = r7212170 * r7212164;
        double r7212172 = r7212167 - r7212171;
        return r7212172;
}

↓

double f(double x_re, double x_im) {
        double r7212173 = x_im;
        double r7212174 = x_re;
        double r7212175 = r7212173 + r7212174;
        double r7212176 = r7212174 - r7212173;
        double r7212177 = r7212176 * r7212174;
        double r7212178 = r7212174 * r7212173;
        double r7212179 = r7212178 + r7212178;
        double r7212180 = cbrt(r7212179);
        double r7212181 = -r7212180;
        double r7212182 = r7212180 * r7212181;
        double r7212183 = r7212180 * r7212182;
        double r7212184 = r7212173 * r7212183;
        double r7212185 = fma(r7212175, r7212177, r7212184);
        return r7212185;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.6
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 6.6

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

3. Applied difference-of-squares 6.6

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

4. Applied associate-*l* 0.2

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

6. Applied fma-neg 0.2

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

7. Simplified 0.2

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

9. Applied add-cube-cbrt 0.6

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

10. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019164 +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)))