math.cube on complex, imaginary part

Average Error: 6.6 → 0.6

Time: 22.9s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r7986145 = x_re;
        double r7986146 = r7986145 * r7986145;
        double r7986147 = x_im;
        double r7986148 = r7986147 * r7986147;
        double r7986149 = r7986146 - r7986148;
        double r7986150 = r7986149 * r7986147;
        double r7986151 = r7986145 * r7986147;
        double r7986152 = r7986147 * r7986145;
        double r7986153 = r7986151 + r7986152;
        double r7986154 = r7986153 * r7986145;
        double r7986155 = r7986150 + r7986154;
        return r7986155;
}

↓

double f(double x_re, double x_im) {
        double r7986156 = x_re;
        double r7986157 = x_im;
        double r7986158 = r7986156 - r7986157;
        double r7986159 = r7986158 * r7986157;
        double r7986160 = r7986157 + r7986156;
        double r7986161 = r7986159 * r7986160;
        double r7986162 = r7986156 * r7986157;
        double r7986163 = r7986162 + r7986162;
        double r7986164 = cbrt(r7986163);
        double r7986165 = r7986164 * r7986164;
        double r7986166 = r7986164 * r7986156;
        double r7986167 = r7986165 * r7986166;
        double r7986168 = r7986161 + r7986167;
        return r7986168;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.6
Target	0.2
Herbie	0.6

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

Derivation

1. Initial program 6.6

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

3. Applied difference-of-squares 6.6

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

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. Using strategy rm

6. Applied add-cube-cbrt 0.6

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

7. Applied associate-*l* 0.6

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

8. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019162 +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)))