math.cube on complex, real part

Average Error: 7.4 → 0.6

Time: 20.3s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r10300224 = x_re;
        double r10300225 = r10300224 * r10300224;
        double r10300226 = x_im;
        double r10300227 = r10300226 * r10300226;
        double r10300228 = r10300225 - r10300227;
        double r10300229 = r10300228 * r10300224;
        double r10300230 = r10300224 * r10300226;
        double r10300231 = r10300226 * r10300224;
        double r10300232 = r10300230 + r10300231;
        double r10300233 = r10300232 * r10300226;
        double r10300234 = r10300229 - r10300233;
        return r10300234;
}

↓

double f(double x_re, double x_im) {
        double r10300235 = x_re;
        double r10300236 = x_im;
        double r10300237 = r10300235 - r10300236;
        double r10300238 = r10300237 * r10300235;
        double r10300239 = r10300236 + r10300235;
        double r10300240 = r10300238 * r10300239;
        double r10300241 = r10300235 * r10300236;
        double r10300242 = r10300241 + r10300241;
        double r10300243 = cbrt(r10300242);
        double r10300244 = r10300243 * r10300243;
        double r10300245 = r10300236 * r10300243;
        double r10300246 = r10300244 * r10300245;
        double r10300247 = r10300240 - r10300246;
        return r10300247;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
Target	0.2
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 7.4

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

3. Applied difference-of-squares 7.4

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

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 add-cube-cbrt 0.6

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

7. Applied associate-*l* 0.6

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

8. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019179 
(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 (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))