2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 16.4s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r4759956 = 2.0;
        double r4759957 = atan2(1.0, 0.0);
        double r4759958 = r4759956 * r4759957;
        double r4759959 = 3.0;
        double r4759960 = r4759958 / r4759959;
        double r4759961 = g;
        double r4759962 = -r4759961;
        double r4759963 = h;
        double r4759964 = r4759962 / r4759963;
        double r4759965 = acos(r4759964);
        double r4759966 = r4759965 / r4759959;
        double r4759967 = r4759960 + r4759966;
        double r4759968 = cos(r4759967);
        double r4759969 = r4759956 * r4759968;
        return r4759969;
}

↓

double f(double g, double h) {
        double r4759970 = 2.0;
        double r4759971 = g;
        double r4759972 = h;
        double r4759973 = r4759971 / r4759972;
        double r4759974 = asin(r4759973);
        double r4759975 = 3.0;
        double r4759976 = r4759974 / r4759975;
        double r4759977 = cos(r4759976);
        double r4759978 = atan2(1.0, 0.0);
        double r4759979 = r4759978 / r4759975;
        double r4759980 = r4759979 * r4759970;
        double r4759981 = r4759979 + r4759980;
        double r4759982 = sin(r4759981);
        double r4759983 = r4759978 / r4759970;
        double r4759984 = r4759983 / r4759975;
        double r4759985 = sin(r4759984);
        double r4759986 = r4759982 * r4759985;
        double r4759987 = cos(r4759984);
        double r4759988 = cos(r4759981);
        double r4759989 = r4759987 * r4759988;
        double r4759990 = r4759986 + r4759989;
        double r4759991 = r4759977 * r4759990;
        double r4759992 = sin(r4759976);
        double r4759993 = r4759981 - r4759984;
        double r4759994 = sin(r4759993);
        double r4759995 = r4759992 * r4759994;
        double r4759996 = r4759991 - r4759995;
        double r4759997 = r4759970 * r4759996;
        return r4759997;
}

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. Using strategy rm

3. Applied distribute-frac-neg 1.0

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

4. Applied acos-neg 1.0

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

5. Applied div-sub 1.0

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

6. Applied associate-+r- 1.0

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

7. Simplified 1.0

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

9. Applied acos-asin 1.0

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

10. Applied div-sub 1.0

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

11. Applied associate--r- 1.0

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

12. Applied cos-sum 1.0

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

14. Applied cos-diff 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019134 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))