Average Error: 1.0 → 0.1
Time: 4.6s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
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{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)
double f(double g, double h) {
        double r109676 = 2.0;
        double r109677 = atan2(1.0, 0.0);
        double r109678 = r109676 * r109677;
        double r109679 = 3.0;
        double r109680 = r109678 / r109679;
        double r109681 = g;
        double r109682 = -r109681;
        double r109683 = h;
        double r109684 = r109682 / r109683;
        double r109685 = acos(r109684);
        double r109686 = r109685 / r109679;
        double r109687 = r109680 + r109686;
        double r109688 = cos(r109687);
        double r109689 = r109676 * r109688;
        return r109689;
}
↓
double f(double g, double h) {
        double r109690 = 2.0;
        double r109691 = atan2(1.0, 0.0);
        double r109692 = r109690 * r109691;
        double r109693 = 3.0;
        double r109694 = r109692 / r109693;
        double r109695 = r109691 / r109693;
        double r109696 = r109694 + r109695;
        double r109697 = cos(r109696);
        double r109698 = g;
        double r109699 = h;
        double r109700 = r109698 / r109699;
        double r109701 = acos(r109700);
        double r109702 = r109701 / r109693;
        double r109703 = cos(r109702);
        double r109704 = r109697 * r109703;
        double r109705 = sin(r109696);
        double r109706 = sin(r109702);
        double r109707 = r109705 * r109706;
        double r109708 = r109704 + r109707;
        double r109709 = r109690 * r109708;
        return r109709;
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 distribute-frac-neg1.0

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

  4. Applied acos-neg1.0

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

  5. Applied div-sub1.0

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

  6. Applied associate-+r-1.0

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

  7. Applied cos-diff0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))