2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 25.5s

Precision: 64

Math

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

↓

\[2 \cdot \left(\sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \log \left(e^{\cos \left(\left(\frac{\pi}{6} - \frac{\pi}{\frac{3}{2}}\right) - \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\right)\]

C

double f(double g, double h) {
        double r5454406 = 2.0;
        double r5454407 = atan2(1.0, 0.0);
        double r5454408 = r5454406 * r5454407;
        double r5454409 = 3.0;
        double r5454410 = r5454408 / r5454409;
        double r5454411 = g;
        double r5454412 = -r5454411;
        double r5454413 = h;
        double r5454414 = r5454412 / r5454413;
        double r5454415 = acos(r5454414);
        double r5454416 = r5454415 / r5454409;
        double r5454417 = r5454410 + r5454416;
        double r5454418 = cos(r5454417);
        double r5454419 = r5454406 * r5454418;
        return r5454419;
}

↓

double f(double g, double h) {
        double r5454420 = 2.0;
        double r5454421 = g;
        double r5454422 = h;
        double r5454423 = r5454421 / r5454422;
        double r5454424 = acos(r5454423);
        double r5454425 = 3.0;
        double r5454426 = r5454424 / r5454425;
        double r5454427 = atan2(1.0, 0.0);
        double r5454428 = 1.5;
        double r5454429 = r5454427 / r5454428;
        double r5454430 = r5454426 - r5454429;
        double r5454431 = sin(r5454430);
        double r5454432 = sqrt(r5454425);
        double r5454433 = r5454432 / r5454420;
        double r5454434 = r5454431 * r5454433;
        double r5454435 = 0.5;
        double r5454436 = 6.0;
        double r5454437 = r5454427 / r5454436;
        double r5454438 = r5454437 - r5454429;
        double r5454439 = asin(r5454423);
        double r5454440 = r5454439 / r5454425;
        double r5454441 = r5454438 - r5454440;
        double r5454442 = cos(r5454441);
        double r5454443 = exp(r5454442);
        double r5454444 = log(r5454443);
        double r5454445 = r5454435 * r5454444;
        double r5454446 = r5454434 + r5454445;
        double r5454447 = r5454420 * r5454446;
        return r5454447;
}

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

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

12. Applied acos-asin 0.1

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

13. Applied div-sub 0.1

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

14. Applied associate--l- 0.0

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

16. Applied add-log-exp 0.0

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

17. Simplified 0.0

    \[\leadsto \left(\frac{1}{2} \cdot \log \color{blue}{\left(e^{\cos \left(\left(\frac{\pi}{6} - \frac{\pi}{\frac{3}{2}}\right) - \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]

18. Final simplification 0.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \log \left(e^{\cos \left(\left(\frac{\pi}{6} - \frac{\pi}{\frac{3}{2}}\right) - \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019144 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))