2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 15.1s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r149786 = 2.0;
        double r149787 = atan2(1.0, 0.0);
        double r149788 = r149786 * r149787;
        double r149789 = 3.0;
        double r149790 = r149788 / r149789;
        double r149791 = g;
        double r149792 = -r149791;
        double r149793 = h;
        double r149794 = r149792 / r149793;
        double r149795 = acos(r149794);
        double r149796 = r149795 / r149789;
        double r149797 = r149790 + r149796;
        double r149798 = cos(r149797);
        double r149799 = r149786 * r149798;
        return r149799;
}

↓

double f(double g, double h) {
        double r149800 = 2.0;
        double r149801 = 3.0;
        double r149802 = r149800 / r149801;
        double r149803 = atan2(1.0, 0.0);
        double r149804 = g;
        double r149805 = h;
        double r149806 = r149804 / r149805;
        double r149807 = -r149806;
        double r149808 = acos(r149807);
        double r149809 = r149808 / r149801;
        double r149810 = fma(r149802, r149803, r149809);
        double r149811 = cos(r149810);
        double r149812 = 2.0;
        double r149813 = pow(r149811, r149812);
        double r149814 = cbrt(r149813);
        double r149815 = cbrt(r149811);
        double r149816 = r149814 * r149815;
        double r149817 = r149800 * r149816;
        return r149817;
}

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

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

4. Applied add-cbrt-cube 1.5

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

5. Simplified 1.0

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

7. Applied cube-mult 1.5

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

8. Applied cbrt-prod 0.1

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

9. Simplified 0.1

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

10. Simplified 0.1

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

11. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))