Average Error: 1.0 → 0.0
Time: 3.9s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3}}\right) - \left(\sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)}\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3}}\right)\right)\]
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}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3}}\right) - \left(\sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)}\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt[3]{3}}\right)\right)
double f(double g, double h) {
        double r145687 = 2.0;
        double r145688 = atan2(1.0, 0.0);
        double r145689 = r145687 * r145688;
        double r145690 = 3.0;
        double r145691 = r145689 / r145690;
        double r145692 = g;
        double r145693 = -r145692;
        double r145694 = h;
        double r145695 = r145693 / r145694;
        double r145696 = acos(r145695);
        double r145697 = r145696 / r145690;
        double r145698 = r145691 + r145697;
        double r145699 = cos(r145698);
        double r145700 = r145687 * r145699;
        return r145700;
}
↓
double f(double g, double h) {
        double r145701 = 2.0;
        double r145702 = atan2(1.0, 0.0);
        double r145703 = r145701 * r145702;
        double r145704 = 3.0;
        double r145705 = r145703 / r145704;
        double r145706 = cos(r145705);
        double r145707 = g;
        double r145708 = -r145707;
        double r145709 = h;
        double r145710 = r145708 / r145709;
        double r145711 = acos(r145710);
        double r145712 = sqrt(r145711);
        double r145713 = cbrt(r145704);
        double r145714 = r145713 * r145713;
        double r145715 = r145712 / r145714;
        double r145716 = r145712 / r145713;
        double r145717 = r145715 * r145716;
        double r145718 = cos(r145717);
        double r145719 = r145706 * r145718;
        double r145720 = sin(r145705);
        double r145721 = sqrt(r145720);
        double r145722 = r145721 * r145721;
        double r145723 = sin(r145717);
        double r145724 = r145722 * r145723;
        double r145725 = r145719 - r145724;
        double r145726 = r145701 * r145725;
        return r145726;
}

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 add-cube-cbrt1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied times-frac1.0

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

  7. Applied cos-sum1.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))