2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 26.0s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r17789648 = 2.0;
        double r17789649 = atan2(1.0, 0.0);
        double r17789650 = r17789648 * r17789649;
        double r17789651 = 3.0;
        double r17789652 = r17789650 / r17789651;
        double r17789653 = g;
        double r17789654 = -r17789653;
        double r17789655 = h;
        double r17789656 = r17789654 / r17789655;
        double r17789657 = acos(r17789656);
        double r17789658 = r17789657 / r17789651;
        double r17789659 = r17789652 + r17789658;
        double r17789660 = cos(r17789659);
        double r17789661 = r17789648 * r17789660;
        return r17789661;
}

↓

double f(double g, double h) {
        double r17789662 = 0.6666666666666666;
        double r17789663 = atan2(1.0, 0.0);
        double r17789664 = g;
        double r17789665 = h;
        double r17789666 = r17789664 / r17789665;
        double r17789667 = -r17789666;
        double r17789668 = acos(r17789667);
        double r17789669 = 3.0;
        double r17789670 = r17789668 / r17789669;
        double r17789671 = fma(r17789662, r17789663, r17789670);
        double r17789672 = cos(r17789671);
        double r17789673 = r17789672 * r17789672;
        double r17789674 = r17789672 * r17789673;
        double r17789675 = cbrt(r17789674);
        double r17789676 = cbrt(r17789675);
        double r17789677 = cbrt(r17789673);
        double r17789678 = r17789676 * r17789677;
        double r17789679 = 2.0;
        double r17789680 = r17789678 * r17789679;
        return r17789680;
}

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 cbrt-unprod 0.1

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

8. Applied add-cbrt-cube 0.1

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

9. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))