Average Error: 1.0 → 0.0
Time: 19.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\left(\sqrt{\pi} \cdot \frac{2}{3}\right) \cdot \sqrt{\pi}\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right) \cdot 2\]
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{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\left(\sqrt{\pi} \cdot \frac{2}{3}\right) \cdot \sqrt{\pi}\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right) \cdot 2
double f(double g, double h) {
        double r5268866 = 2.0;
        double r5268867 = atan2(1.0, 0.0);
        double r5268868 = r5268866 * r5268867;
        double r5268869 = 3.0;
        double r5268870 = r5268868 / r5268869;
        double r5268871 = g;
        double r5268872 = -r5268871;
        double r5268873 = h;
        double r5268874 = r5268872 / r5268873;
        double r5268875 = acos(r5268874);
        double r5268876 = r5268875 / r5268869;
        double r5268877 = r5268870 + r5268876;
        double r5268878 = cos(r5268877);
        double r5268879 = r5268866 * r5268878;
        return r5268879;
}
↓
double f(double g, double h) {
        double r5268880 = 0.6666666666666666;
        double r5268881 = atan2(1.0, 0.0);
        double r5268882 = r5268880 * r5268881;
        double r5268883 = cos(r5268882);
        double r5268884 = g;
        double r5268885 = h;
        double r5268886 = r5268884 / r5268885;
        double r5268887 = -r5268886;
        double r5268888 = acos(r5268887);
        double r5268889 = 3.0;
        double r5268890 = r5268888 / r5268889;
        double r5268891 = cos(r5268890);
        double r5268892 = r5268883 * r5268891;
        double r5268893 = sqrt(r5268881);
        double r5268894 = r5268893 * r5268880;
        double r5268895 = r5268894 * r5268893;
        double r5268896 = sin(r5268895);
        double r5268897 = sin(r5268890);
        double r5268898 = r5268896 * r5268897;
        double r5268899 = r5268892 - r5268898;
        double r5268900 = 2.0;
        double r5268901 = r5268899 * r5268900;
        return r5268901;
}

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 fma-udef1.0

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

  5. Applied cos-sum1.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied associate-*r*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))