2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 19.3s

Precision: 64

Math

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

↓

\[\left(\cos \left(\frac{\frac{\pi}{2}}{3}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right) + \sin \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right) \cdot \sin \left(\frac{\sqrt{\pi}}{6} \cdot \sqrt{\pi}\right)\right) \cdot 2\]

C

double f(double g, double h) {
        double r4440718 = 2.0;
        double r4440719 = atan2(1.0, 0.0);
        double r4440720 = r4440718 * r4440719;
        double r4440721 = 3.0;
        double r4440722 = r4440720 / r4440721;
        double r4440723 = g;
        double r4440724 = -r4440723;
        double r4440725 = h;
        double r4440726 = r4440724 / r4440725;
        double r4440727 = acos(r4440726);
        double r4440728 = r4440727 / r4440721;
        double r4440729 = r4440722 + r4440728;
        double r4440730 = cos(r4440729);
        double r4440731 = r4440718 * r4440730;
        return r4440731;
}

↓

double f(double g, double h) {
        double r4440732 = atan2(1.0, 0.0);
        double r4440733 = 2.0;
        double r4440734 = r4440732 / r4440733;
        double r4440735 = 3.0;
        double r4440736 = r4440734 / r4440735;
        double r4440737 = cos(r4440736);
        double r4440738 = g;
        double r4440739 = -r4440738;
        double r4440740 = h;
        double r4440741 = r4440739 / r4440740;
        double r4440742 = asin(r4440741);
        double r4440743 = r4440742 / r4440735;
        double r4440744 = 1.5;
        double r4440745 = sqrt(r4440744);
        double r4440746 = r4440732 / r4440745;
        double r4440747 = r4440746 / r4440745;
        double r4440748 = r4440743 - r4440747;
        double r4440749 = cos(r4440748);
        double r4440750 = r4440737 * r4440749;
        double r4440751 = sin(r4440748);
        double r4440752 = sqrt(r4440732);
        double r4440753 = 6.0;
        double r4440754 = r4440752 / r4440753;
        double r4440755 = r4440754 * r4440752;
        double r4440756 = sin(r4440755);
        double r4440757 = r4440751 * r4440756;
        double r4440758 = r4440750 + r4440757;
        double r4440759 = r4440758 * r4440733;
        return r4440759;
}

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

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

4. Applied add-sqr-sqrt 1.0

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

5. Applied associate-/r* 1.0

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

7. Applied acos-asin 1.0

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

8. Applied div-sub 1.0

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

9. Applied associate-+l- 1.0

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

10. Applied cos-diff 1.0

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

12. Applied *-un-lft-identity 1.0

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

13. Applied *-un-lft-identity 1.0

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

14. Applied add-sqr-sqrt 1.0

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

15. Applied times-frac 1.0

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

16. Applied times-frac 0.0

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

17. Simplified 0.0

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

18. Simplified 0.0

    \[\leadsto \left(\cos \left(\frac{\frac{\pi}{2}}{3}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right) + \sin \left(\sqrt{\pi} \cdot \color{blue}{\frac{\sqrt{\pi}}{6}}\right) \cdot \sin \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right)\right) \cdot 2\]

19. Final simplification 0.0

    \[\leadsto \left(\cos \left(\frac{\frac{\pi}{2}}{3}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right) + \sin \left(\frac{\sin^{-1} \left(\frac{-g}{h}\right)}{3} - \frac{\frac{\pi}{\sqrt{\frac{3}{2}}}}{\sqrt{\frac{3}{2}}}\right) \cdot \sin \left(\frac{\sqrt{\pi}}{6} \cdot \sqrt{\pi}\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019130 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))