2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 17.9s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r4714296 = 2.0;
        double r4714297 = atan2(1.0, 0.0);
        double r4714298 = r4714296 * r4714297;
        double r4714299 = 3.0;
        double r4714300 = r4714298 / r4714299;
        double r4714301 = g;
        double r4714302 = -r4714301;
        double r4714303 = h;
        double r4714304 = r4714302 / r4714303;
        double r4714305 = acos(r4714304);
        double r4714306 = r4714305 / r4714299;
        double r4714307 = r4714300 + r4714306;
        double r4714308 = cos(r4714307);
        double r4714309 = r4714296 * r4714308;
        return r4714309;
}

↓

double f(double g, double h) {
        double r4714310 = 2.0;
        double r4714311 = atan2(1.0, 0.0);
        double r4714312 = r4714311 / r4714310;
        double r4714313 = 3.0;
        double r4714314 = r4714312 / r4714313;
        double r4714315 = 1.5;
        double r4714316 = r4714311 / r4714315;
        double r4714317 = g;
        double r4714318 = h;
        double r4714319 = r4714317 / r4714318;
        double r4714320 = asin(r4714319);
        double r4714321 = r4714320 / r4714313;
        double r4714322 = r4714316 + r4714321;
        double r4714323 = r4714314 - r4714322;
        double r4714324 = cos(r4714323);
        double r4714325 = 0.5;
        double r4714326 = r4714324 * r4714325;
        double r4714327 = acos(r4714319);
        double r4714328 = r4714327 / r4714313;
        double r4714329 = r4714328 - r4714316;
        double r4714330 = sin(r4714329);
        double r4714331 = r4714330 / r4714310;
        double r4714332 = sqrt(r4714313);
        double r4714333 = r4714331 * r4714332;
        double r4714334 = r4714326 + r4714333;
        double r4714335 = r4714310 * r4714334;
        return r4714335;
}

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

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

10. Simplified 0.1

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

12. Applied acos-asin 0.1

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

13. Applied div-sub 0.1

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

14. Applied associate--l- 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019135 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))