2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 14.1s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r2163623 = 2.0;
        double r2163624 = atan2(1.0, 0.0);
        double r2163625 = r2163623 * r2163624;
        double r2163626 = 3.0;
        double r2163627 = r2163625 / r2163626;
        double r2163628 = g;
        double r2163629 = -r2163628;
        double r2163630 = h;
        double r2163631 = r2163629 / r2163630;
        double r2163632 = acos(r2163631);
        double r2163633 = r2163632 / r2163626;
        double r2163634 = r2163627 + r2163633;
        double r2163635 = cos(r2163634);
        double r2163636 = r2163623 * r2163635;
        return r2163636;
}

↓

double f(double g, double h) {
        double r2163637 = 2.0;
        double r2163638 = 0.6666666666666666;
        double r2163639 = atan2(1.0, 0.0);
        double r2163640 = g;
        double r2163641 = -r2163640;
        double r2163642 = h;
        double r2163643 = r2163641 / r2163642;
        double r2163644 = acos(r2163643);
        double r2163645 = 3.0;
        double r2163646 = r2163644 / r2163645;
        double r2163647 = fma(r2163638, r2163639, r2163646);
        double r2163648 = cos(r2163647);
        double r2163649 = cbrt(r2163648);
        double r2163650 = r2163648 * r2163648;
        double r2163651 = cbrt(r2163650);
        double r2163652 = r2163649 * r2163651;
        double r2163653 = r2163637 * r2163652;
        return r2163653;
}

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-cube-cbrt 1.0

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

6. Applied cbrt-unprod 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))