math.cube on complex, imaginary part

Average Error: 7.1 → 0.7

Time: 22.8s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8155073 = x_re;
        double r8155074 = r8155073 * r8155073;
        double r8155075 = x_im;
        double r8155076 = r8155075 * r8155075;
        double r8155077 = r8155074 - r8155076;
        double r8155078 = r8155077 * r8155075;
        double r8155079 = r8155073 * r8155075;
        double r8155080 = r8155075 * r8155073;
        double r8155081 = r8155079 + r8155080;
        double r8155082 = r8155081 * r8155073;
        double r8155083 = r8155078 + r8155082;
        return r8155083;
}

↓

double f(double x_re, double x_im) {
        double r8155084 = x_im;
        double r8155085 = x_re;
        double r8155086 = r8155084 + r8155085;
        double r8155087 = r8155085 - r8155084;
        double r8155088 = r8155084 * r8155087;
        double r8155089 = r8155086 * r8155088;
        double r8155090 = cbrt(r8155089);
        double r8155091 = cbrt(r8155086);
        double r8155092 = cbrt(r8155088);
        double r8155093 = r8155091 * r8155092;
        double r8155094 = r8155090 * r8155093;
        double r8155095 = r8155090 * r8155094;
        double r8155096 = r8155085 * r8155084;
        double r8155097 = r8155096 + r8155096;
        double r8155098 = r8155085 * r8155097;
        double r8155099 = r8155095 + r8155098;
        return r8155099;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.2
Herbie	0.7

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

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* 0.2

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

6. Applied add-cube-cbrt 0.7

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

8. Applied cbrt-prod 0.7

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

9. Final simplification 0.7

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))