2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 3.8s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r220425 = 2.0;
        double r220426 = atan2(1.0, 0.0);
        double r220427 = r220425 * r220426;
        double r220428 = 3.0;
        double r220429 = r220427 / r220428;
        double r220430 = g;
        double r220431 = -r220430;
        double r220432 = h;
        double r220433 = r220431 / r220432;
        double r220434 = acos(r220433);
        double r220435 = r220434 / r220428;
        double r220436 = r220429 + r220435;
        double r220437 = cos(r220436);
        double r220438 = r220425 * r220437;
        return r220438;
}

↓

double f(double g, double h) {
        double r220439 = 2.0;
        double r220440 = atan2(1.0, 0.0);
        double r220441 = r220439 * r220440;
        double r220442 = 3.0;
        double r220443 = r220441 / r220442;
        double r220444 = g;
        double r220445 = -r220444;
        double r220446 = h;
        double r220447 = r220445 / r220446;
        double r220448 = acos(r220447);
        double r220449 = r220448 / r220442;
        double r220450 = r220443 + r220449;
        double r220451 = cos(r220450);
        double r220452 = cbrt(r220443);
        double r220453 = r220452 * r220452;
        double r220454 = r220453 * r220452;
        double r220455 = r220454 + r220449;
        double r220456 = cos(r220455);
        double r220457 = r220451 * r220456;
        double r220458 = cbrt(r220457);
        double r220459 = cbrt(r220451);
        double r220460 = r220458 * r220459;
        double r220461 = r220439 * r220460;
        return r220461;
}

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 add-cbrt-cube 1.6

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

4. Simplified 1.0

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

6. Applied add-cube-cbrt 1.0

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

7. Applied unpow-prod-down 1.0

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

8. Applied cbrt-prod 0.1

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

9. Simplified 0.1

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

10. Simplified 0.1

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

12. Applied add-cube-cbrt 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020020 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))