2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 32.1s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r29497221 = 2.0;
        double r29497222 = atan2(1.0, 0.0);
        double r29497223 = r29497221 * r29497222;
        double r29497224 = 3.0;
        double r29497225 = r29497223 / r29497224;
        double r29497226 = g;
        double r29497227 = -r29497226;
        double r29497228 = h;
        double r29497229 = r29497227 / r29497228;
        double r29497230 = acos(r29497229);
        double r29497231 = r29497230 / r29497224;
        double r29497232 = r29497225 + r29497231;
        double r29497233 = cos(r29497232);
        double r29497234 = r29497221 * r29497233;
        return r29497234;
}

↓

double f(double g, double h) {
        double r29497235 = 2.0;
        double r29497236 = 0.5;
        double r29497237 = atan2(1.0, 0.0);
        double r29497238 = r29497237 / r29497235;
        double r29497239 = 3.0;
        double r29497240 = r29497238 / r29497239;
        double r29497241 = 1.5;
        double r29497242 = r29497237 / r29497241;
        double r29497243 = g;
        double r29497244 = h;
        double r29497245 = r29497243 / r29497244;
        double r29497246 = asin(r29497245);
        double r29497247 = r29497246 / r29497239;
        double r29497248 = r29497242 + r29497247;
        double r29497249 = r29497240 - r29497248;
        double r29497250 = cos(r29497249);
        double r29497251 = r29497236 * r29497250;
        double r29497252 = sqrt(r29497239);
        double r29497253 = r29497252 / r29497235;
        double r29497254 = acos(r29497245);
        double r29497255 = r29497254 / r29497239;
        double r29497256 = r29497255 - r29497242;
        double r29497257 = sin(r29497256);
        double r29497258 = r29497253 * r29497257;
        double r29497259 = r29497251 + r29497258;
        double r29497260 = r29497235 * r29497259;
        return r29497260;
}

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. Simplified 0.1

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

12. Applied acos-asin 0.1

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

13. Applied div-sub 0.1

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

14. Applied associate--l- 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019125 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))