2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 16.9s

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 r18369536 = 2.0;
        double r18369537 = atan2(1.0, 0.0);
        double r18369538 = r18369536 * r18369537;
        double r18369539 = 3.0;
        double r18369540 = r18369538 / r18369539;
        double r18369541 = g;
        double r18369542 = -r18369541;
        double r18369543 = h;
        double r18369544 = r18369542 / r18369543;
        double r18369545 = acos(r18369544);
        double r18369546 = r18369545 / r18369539;
        double r18369547 = r18369540 + r18369546;
        double r18369548 = cos(r18369547);
        double r18369549 = r18369536 * r18369548;
        return r18369549;
}

↓

double f(double g, double h) {
        double r18369550 = atan2(1.0, 0.0);
        double r18369551 = 1.5;
        double r18369552 = r18369550 / r18369551;
        double r18369553 = cos(r18369552);
        double r18369554 = g;
        double r18369555 = h;
        double r18369556 = r18369554 / r18369555;
        double r18369557 = acos(r18369556);
        double r18369558 = 3.0;
        double r18369559 = r18369557 / r18369558;
        double r18369560 = cos(r18369559);
        double r18369561 = r18369553 * r18369560;
        double r18369562 = sin(r18369552);
        double r18369563 = sin(r18369559);
        double r18369564 = r18369562 * r18369563;
        double r18369565 = r18369561 + r18369564;
        double r18369566 = 0.5;
        double r18369567 = r18369565 * r18369566;
        double r18369568 = r18369559 - r18369552;
        double r18369569 = sin(r18369568);
        double r18369570 = r18369550 / r18369558;
        double r18369571 = sqrt(r18369570);
        double r18369572 = r18369571 * r18369571;
        double r18369573 = sin(r18369572);
        double r18369574 = r18369569 * r18369573;
        double r18369575 = r18369567 + r18369574;
        double r18369576 = 2.0;
        double r18369577 = r18369575 * r18369576;
        return r18369577;
}

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 2019119 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))