2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 16.9s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r5166413 = 2.0;
        double r5166414 = atan2(1.0, 0.0);
        double r5166415 = r5166413 * r5166414;
        double r5166416 = 3.0;
        double r5166417 = r5166415 / r5166416;
        double r5166418 = g;
        double r5166419 = -r5166418;
        double r5166420 = h;
        double r5166421 = r5166419 / r5166420;
        double r5166422 = acos(r5166421);
        double r5166423 = r5166422 / r5166416;
        double r5166424 = r5166417 + r5166423;
        double r5166425 = cos(r5166424);
        double r5166426 = r5166413 * r5166425;
        return r5166426;
}

↓

double f(double g, double h) {
        double r5166427 = 2.0;
        double r5166428 = atan2(1.0, 0.0);
        double r5166429 = 3.0;
        double r5166430 = r5166428 / r5166429;
        double r5166431 = r5166427 * r5166428;
        double r5166432 = r5166431 / r5166429;
        double r5166433 = r5166430 + r5166432;
        double r5166434 = cos(r5166433);
        double r5166435 = g;
        double r5166436 = h;
        double r5166437 = r5166435 / r5166436;
        double r5166438 = acos(r5166437);
        double r5166439 = r5166438 / r5166429;
        double r5166440 = cos(r5166439);
        double r5166441 = r5166434 * r5166440;
        double r5166442 = sin(r5166433);
        double r5166443 = sin(r5166439);
        double r5166444 = r5166442 * r5166443;
        double r5166445 = r5166441 + r5166444;
        double r5166446 = r5166427 * r5166445;
        return r5166446;
}

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. Applied cos-diff 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019179 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))