math.cube on complex, imaginary part

Average Error: 7.0 → 0.6

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r48495049 = x_re;
        double r48495050 = r48495049 * r48495049;
        double r48495051 = x_im;
        double r48495052 = r48495051 * r48495051;
        double r48495053 = r48495050 - r48495052;
        double r48495054 = r48495053 * r48495051;
        double r48495055 = r48495049 * r48495051;
        double r48495056 = r48495051 * r48495049;
        double r48495057 = r48495055 + r48495056;
        double r48495058 = r48495057 * r48495049;
        double r48495059 = r48495054 + r48495058;
        return r48495059;
}

↓

double f(double x_re, double x_im) {
        double r48495060 = x_im;
        double r48495061 = x_re;
        double r48495062 = r48495060 + r48495061;
        double r48495063 = r48495062 * r48495060;
        double r48495064 = r48495061 - r48495060;
        double r48495065 = r48495063 * r48495064;
        double r48495066 = r48495061 * r48495060;
        double r48495067 = r48495066 + r48495066;
        double r48495068 = cbrt(r48495067);
        double r48495069 = r48495068 * r48495068;
        double r48495070 = r48495068 * r48495061;
        double r48495071 = r48495069 * r48495070;
        double r48495072 = r48495065 + r48495071;
        return r48495072;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
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 7.0

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

2. Taylor expanded around inf 6.9

   \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

3. Simplified 0.2

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

5. Applied add-cube-cbrt 0.6

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

6. Applied associate-*l* 0.6

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

7. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019128 
(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)))