2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 3.9s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r98748 = 2.0;
        double r98749 = atan2(1.0, 0.0);
        double r98750 = r98748 * r98749;
        double r98751 = 3.0;
        double r98752 = r98750 / r98751;
        double r98753 = g;
        double r98754 = -r98753;
        double r98755 = h;
        double r98756 = r98754 / r98755;
        double r98757 = acos(r98756);
        double r98758 = r98757 / r98751;
        double r98759 = r98752 + r98758;
        double r98760 = cos(r98759);
        double r98761 = r98748 * r98760;
        return r98761;
}

↓

double f(double g, double h) {
        double r98762 = 2.0;
        double r98763 = atan2(1.0, 0.0);
        double r98764 = r98762 * r98763;
        double r98765 = 3.0;
        double r98766 = r98764 / r98765;
        double r98767 = cos(r98766);
        double r98768 = cbrt(r98767);
        double r98769 = r98768 * r98768;
        double r98770 = r98769 * r98768;
        double r98771 = g;
        double r98772 = -r98771;
        double r98773 = h;
        double r98774 = r98772 / r98773;
        double r98775 = acos(r98774);
        double r98776 = r98775 / r98765;
        double r98777 = cos(r98776);
        double r98778 = r98770 * r98777;
        double r98779 = sin(r98766);
        double r98780 = sin(r98776);
        double r98781 = r98779 * r98780;
        double r98782 = r98778 - r98781;
        double r98783 = r98762 * r98782;
        return r98783;
}

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. Using strategy rm

3. Applied cos-sum 1.0

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

5. Applied add-cube-cbrt 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020021 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))