Average Error: 1.0 → 0.0
Time: 8.2s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2 \cdot \pi}{3}\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \left(\sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)}\right)\right)\]
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{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2 \cdot \pi}{3}\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \left(\sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)}\right)\right)
double f(double g, double h) {
        double r129950 = 2.0;
        double r129951 = atan2(1.0, 0.0);
        double r129952 = r129950 * r129951;
        double r129953 = 3.0;
        double r129954 = r129952 / r129953;
        double r129955 = g;
        double r129956 = -r129955;
        double r129957 = h;
        double r129958 = r129956 / r129957;
        double r129959 = acos(r129958);
        double r129960 = r129959 / r129953;
        double r129961 = r129954 + r129960;
        double r129962 = cos(r129961);
        double r129963 = r129950 * r129962;
        return r129963;
}
↓
double f(double g, double h) {
        double r129964 = 2.0;
        double r129965 = g;
        double r129966 = -r129965;
        double r129967 = h;
        double r129968 = r129966 / r129967;
        double r129969 = acos(r129968);
        double r129970 = 3.0;
        double r129971 = r129969 / r129970;
        double r129972 = cos(r129971);
        double r129973 = atan2(1.0, 0.0);
        double r129974 = r129964 * r129973;
        double r129975 = r129974 / r129970;
        double r129976 = cos(r129975);
        double r129977 = r129972 * r129976;
        double r129978 = sin(r129971);
        double r129979 = sin(r129975);
        double r129980 = sqrt(r129979);
        double r129981 = r129980 * r129980;
        double r129982 = r129978 * r129981;
        double r129983 = r129977 - r129982;
        double r129984 = r129964 * r129983;
        return r129984;
}

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 associate-/r*1.0

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

  6. Applied cos-sum1.0

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

  7. Simplified0.0

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

  8. Simplified1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))