2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 19.6s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r4814568 = 2.0;
        double r4814569 = atan2(1.0, 0.0);
        double r4814570 = r4814568 * r4814569;
        double r4814571 = 3.0;
        double r4814572 = r4814570 / r4814571;
        double r4814573 = g;
        double r4814574 = -r4814573;
        double r4814575 = h;
        double r4814576 = r4814574 / r4814575;
        double r4814577 = acos(r4814576);
        double r4814578 = r4814577 / r4814571;
        double r4814579 = r4814572 + r4814578;
        double r4814580 = cos(r4814579);
        double r4814581 = r4814568 * r4814580;
        return r4814581;
}

↓

double f(double g, double h) {
        double r4814582 = 2.0;
        double r4814583 = g;
        double r4814584 = h;
        double r4814585 = r4814583 / r4814584;
        double r4814586 = acos(r4814585);
        double r4814587 = 3.0;
        double r4814588 = r4814586 / r4814587;
        double r4814589 = atan2(1.0, 0.0);
        double r4814590 = 1.5;
        double r4814591 = r4814589 / r4814590;
        double r4814592 = r4814588 - r4814591;
        double r4814593 = sin(r4814592);
        double r4814594 = sqrt(r4814587);
        double r4814595 = r4814594 / r4814582;
        double r4814596 = r4814593 * r4814595;
        double r4814597 = 6.0;
        double r4814598 = r4814589 / r4814597;
        double r4814599 = r4814598 - r4814591;
        double r4814600 = asin(r4814585);
        double r4814601 = r4814600 / r4814587;
        double r4814602 = r4814599 - r4814601;
        double r4814603 = cos(r4814602);
        double r4814604 = exp(r4814603);
        double r4814605 = log(r4814604);
        double r4814606 = 0.5;
        double r4814607 = r4814605 * r4814606;
        double r4814608 = r4814596 + r4814607;
        double r4814609 = r4814582 * r4814608;
        return r4814609;
}

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

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

12. Applied acos-asin 0.1

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

13. Applied div-sub 0.0

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

14. Applied associate--l- 0.0

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

16. Applied add-log-exp 0.0

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

17. Simplified 0.0

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

18. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019139 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))