Average Error: 1.0 → 0.0
Time: 4.2s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]
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}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)
double f(double g, double h) {
        double r169768 = 2.0;
        double r169769 = atan2(1.0, 0.0);
        double r169770 = r169768 * r169769;
        double r169771 = 3.0;
        double r169772 = r169770 / r169771;
        double r169773 = g;
        double r169774 = -r169773;
        double r169775 = h;
        double r169776 = r169774 / r169775;
        double r169777 = acos(r169776);
        double r169778 = r169777 / r169771;
        double r169779 = r169772 + r169778;
        double r169780 = cos(r169779);
        double r169781 = r169768 * r169780;
        return r169781;
}
↓
double f(double g, double h) {
        double r169782 = 2.0;
        double r169783 = atan2(1.0, 0.0);
        double r169784 = r169782 * r169783;
        double r169785 = 3.0;
        double r169786 = r169784 / r169785;
        double r169787 = cos(r169786);
        double r169788 = g;
        double r169789 = -r169788;
        double r169790 = h;
        double r169791 = r169789 / r169790;
        double r169792 = acos(r169791);
        double r169793 = sqrt(r169792);
        double r169794 = sqrt(r169785);
        double r169795 = r169793 / r169794;
        double r169796 = r169795 * r169795;
        double r169797 = cos(r169796);
        double r169798 = r169787 * r169797;
        double r169799 = log1p(r169786);
        double r169800 = expm1(r169799);
        double r169801 = sin(r169800);
        double r169802 = sin(r169796);
        double r169803 = r169801 * r169802;
        double r169804 = r169798 - r169803;
        double r169805 = r169782 * r169804;
        return r169805;
}

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-sqr-sqrt1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied times-frac1.0

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

  7. Applied cos-sum1.0

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

  9. Applied expm1-log1p-u0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right)} \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]

  10. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))