Average Error: 1.0 → 0.0
Time: 25.0s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) \cdot \left(\sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt{\sin \left(\frac{2}{3} \cdot \pi\right)}\right)\right) \cdot 2
double f(double g, double h) {
        double r5500826 = 2.0;
        double r5500827 = atan2(1.0, 0.0);
        double r5500828 = r5500826 * r5500827;
        double r5500829 = 3.0;
        double r5500830 = r5500828 / r5500829;
        double r5500831 = g;
        double r5500832 = -r5500831;
        double r5500833 = h;
        double r5500834 = r5500832 / r5500833;
        double r5500835 = acos(r5500834);
        double r5500836 = r5500835 / r5500829;
        double r5500837 = r5500830 + r5500836;
        double r5500838 = cos(r5500837);
        double r5500839 = r5500826 * r5500838;
        return r5500839;
}
↓
double f(double g, double h) {
        double r5500840 = 0.6666666666666666;
        double r5500841 = atan2(1.0, 0.0);
        double r5500842 = r5500840 * r5500841;
        double r5500843 = cos(r5500842);
        double r5500844 = g;
        double r5500845 = h;
        double r5500846 = r5500844 / r5500845;
        double r5500847 = -r5500846;
        double r5500848 = acos(r5500847);
        double r5500849 = 3.0;
        double r5500850 = sqrt(r5500849);
        double r5500851 = r5500848 / r5500850;
        double r5500852 = r5500851 / r5500850;
        double r5500853 = cos(r5500852);
        double r5500854 = r5500843 * r5500853;
        double r5500855 = sin(r5500852);
        double r5500856 = sin(r5500842);
        double r5500857 = sqrt(r5500856);
        double r5500858 = r5500857 * r5500857;
        double r5500859 = r5500855 * r5500858;
        double r5500860 = r5500854 - r5500859;
        double r5500861 = 2.0;
        double r5500862 = r5500860 * r5500861;
        return r5500862;
}

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. Simplified1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied associate-/r*1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))