2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 17.3s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r19565456 = 2.0;
        double r19565457 = atan2(1.0, 0.0);
        double r19565458 = r19565456 * r19565457;
        double r19565459 = 3.0;
        double r19565460 = r19565458 / r19565459;
        double r19565461 = g;
        double r19565462 = -r19565461;
        double r19565463 = h;
        double r19565464 = r19565462 / r19565463;
        double r19565465 = acos(r19565464);
        double r19565466 = r19565465 / r19565459;
        double r19565467 = r19565460 + r19565466;
        double r19565468 = cos(r19565467);
        double r19565469 = r19565456 * r19565468;
        return r19565469;
}

↓

double f(double g, double h) {
        double r19565470 = atan2(1.0, 0.0);
        double r19565471 = 1.5;
        double r19565472 = r19565470 / r19565471;
        double r19565473 = cos(r19565472);
        double r19565474 = g;
        double r19565475 = h;
        double r19565476 = r19565474 / r19565475;
        double r19565477 = acos(r19565476);
        double r19565478 = 3.0;
        double r19565479 = r19565477 / r19565478;
        double r19565480 = cos(r19565479);
        double r19565481 = r19565473 * r19565480;
        double r19565482 = sin(r19565472);
        double r19565483 = sin(r19565479);
        double r19565484 = r19565482 * r19565483;
        double r19565485 = r19565481 + r19565484;
        double r19565486 = 0.5;
        double r19565487 = r19565485 * r19565486;
        double r19565488 = r19565479 - r19565472;
        double r19565489 = sin(r19565488);
        double r19565490 = r19565470 / r19565478;
        double r19565491 = sqrt(r19565490);
        double r19565492 = r19565491 * r19565491;
        double r19565493 = sin(r19565492);
        double r19565494 = r19565489 * r19565493;
        double r19565495 = r19565487 + r19565494;
        double r19565496 = 2.0;
        double r19565497 = r19565495 * r19565496;
        return r19565497;
}

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 distribute-frac-neg 1.0

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

5. Applied acos-neg 1.0

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

6. Applied div-sub 1.0

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

7. Applied associate-+l- 1.0

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

8. Applied cos-diff 0.1

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

9. Simplified 0.1

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

11. Applied add-sqr-sqrt 1.0

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

13. Applied cos-diff 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019107 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))