2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 30.8s

Precision: 64

Math

\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

↓

\[\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right)\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right)}\right) \cdot 2\]

C

double f(double g, double h) {
        double r20516883 = 2.0;
        double r20516884 = atan2(1.0, 0.0);
        double r20516885 = r20516883 * r20516884;
        double r20516886 = 3.0;
        double r20516887 = r20516885 / r20516886;
        double r20516888 = g;
        double r20516889 = -r20516888;
        double r20516890 = h;
        double r20516891 = r20516889 / r20516890;
        double r20516892 = acos(r20516891);
        double r20516893 = r20516892 / r20516886;
        double r20516894 = r20516887 + r20516893;
        double r20516895 = cos(r20516894);
        double r20516896 = r20516883 * r20516895;
        return r20516896;
}

↓

double f(double g, double h) {
        double r20516897 = 0.6666666666666666;
        double r20516898 = atan2(1.0, 0.0);
        double r20516899 = g;
        double r20516900 = h;
        double r20516901 = r20516899 / r20516900;
        double r20516902 = -r20516901;
        double r20516903 = acos(r20516902);
        double r20516904 = 3.0;
        double r20516905 = r20516903 / r20516904;
        double r20516906 = fma(r20516897, r20516898, r20516905);
        double r20516907 = cos(r20516906);
        double r20516908 = cbrt(r20516907);
        double r20516909 = r20516908 * r20516908;
        double r20516910 = r20516909 * r20516909;
        double r20516911 = r20516910 * r20516909;
        double r20516912 = cbrt(r20516911);
        double r20516913 = r20516908 * r20516912;
        double r20516914 = 2.0;
        double r20516915 = r20516913 * r20516914;
        return r20516915;
}

Error

Bits error versus g

Bits error versus h

Derivation

1. Initial program 1.0

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

2. Simplified 1.0

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) \cdot 2}\]
3. Using strategy rm

4. Applied add-cube-cbrt 1.0

   \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right)} \cdot 2\]
5. Using strategy rm

6. Applied add-cbrt-cube 0.1

   \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right)\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right)}} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}\right) \cdot 2\]

7. Final simplification 0.1

   \[\leadsto \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right)\right) \cdot \left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\right)}\right)}\right) \cdot 2\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))