math.cube on complex, real part

Average Error: 7.4 → 0.6

Time: 19.7s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8597960 = x_re;
        double r8597961 = r8597960 * r8597960;
        double r8597962 = x_im;
        double r8597963 = r8597962 * r8597962;
        double r8597964 = r8597961 - r8597963;
        double r8597965 = r8597964 * r8597960;
        double r8597966 = r8597960 * r8597962;
        double r8597967 = r8597962 * r8597960;
        double r8597968 = r8597966 + r8597967;
        double r8597969 = r8597968 * r8597962;
        double r8597970 = r8597965 - r8597969;
        return r8597970;
}

↓

double f(double x_re, double x_im) {
        double r8597971 = x_re;
        double r8597972 = x_im;
        double r8597973 = r8597971 - r8597972;
        double r8597974 = r8597973 * r8597971;
        double r8597975 = r8597972 + r8597971;
        double r8597976 = r8597974 * r8597975;
        double r8597977 = r8597971 * r8597972;
        double r8597978 = r8597977 + r8597977;
        double r8597979 = cbrt(r8597978);
        double r8597980 = r8597979 * r8597979;
        double r8597981 = r8597972 * r8597979;
        double r8597982 = r8597980 * r8597981;
        double r8597983 = r8597976 - r8597982;
        return r8597983;
}

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 +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 (* 3.0 x.im))))

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