2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 13.0s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r98250 = 2.0;
        double r98251 = atan2(1.0, 0.0);
        double r98252 = r98250 * r98251;
        double r98253 = 3.0;
        double r98254 = r98252 / r98253;
        double r98255 = g;
        double r98256 = -r98255;
        double r98257 = h;
        double r98258 = r98256 / r98257;
        double r98259 = acos(r98258);
        double r98260 = r98259 / r98253;
        double r98261 = r98254 + r98260;
        double r98262 = cos(r98261);
        double r98263 = r98250 * r98262;
        return r98263;
}

↓

double f(double g, double h) {
        double r98264 = 2.0;
        double r98265 = 3.0;
        double r98266 = g;
        double r98267 = -r98266;
        double r98268 = h;
        double r98269 = r98267 / r98268;
        double r98270 = acos(r98269);
        double r98271 = atan2(1.0, 0.0);
        double r98272 = r98264 * r98271;
        double r98273 = r98270 + r98272;
        double r98274 = r98265 * r98273;
        double r98275 = r98265 * r98265;
        double r98276 = r98274 / r98275;
        double r98277 = cos(r98276);
        double r98278 = r98277 * r98277;
        double r98279 = cbrt(r98278);
        double r98280 = cbrt(r98277);
        double r98281 = r98279 * r98280;
        double r98282 = r98264 * r98281;
        return r98282;
}

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 frac-add 1.0

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

4. Simplified 1.0

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

6. Applied add-cbrt-cube 1.5

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

7. Simplified 1.0

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

9. Applied add-cube-cbrt 1.0

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

10. Applied unpow-prod-down 1.0

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

11. Applied cbrt-prod 0.1

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

12. Simplified 0.1

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

13. Simplified 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))