math.cube on complex, imaginary part

Average Error: 7.1 → 0.8

Time: 17.6s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r3627940 = x_re;
        double r3627941 = r3627940 * r3627940;
        double r3627942 = x_im;
        double r3627943 = r3627942 * r3627942;
        double r3627944 = r3627941 - r3627943;
        double r3627945 = r3627944 * r3627942;
        double r3627946 = r3627940 * r3627942;
        double r3627947 = r3627942 * r3627940;
        double r3627948 = r3627946 + r3627947;
        double r3627949 = r3627948 * r3627940;
        double r3627950 = r3627945 + r3627949;
        return r3627950;
}

↓

double f(double x_re, double x_im) {
        double r3627951 = x_im;
        double r3627952 = x_re;
        double r3627953 = r3627951 + r3627952;
        double r3627954 = r3627952 - r3627951;
        double r3627955 = cbrt(r3627951);
        double r3627956 = r3627955 * r3627955;
        double r3627957 = r3627954 * r3627956;
        double r3627958 = r3627957 * r3627955;
        double r3627959 = r3627952 * r3627951;
        double r3627960 = r3627959 + r3627959;
        double r3627961 = r3627952 * r3627960;
        double r3627962 = fma(r3627953, r3627958, r3627961);
        return r3627962;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.3
Herbie	0.8

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

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* 0.3

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

6. Applied fma-def 0.2

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

8. Applied add-cube-cbrt 0.8

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

9. Applied associate-*r* 0.8

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

10. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019152 +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)))