Average Error: 1.0 → 0.1
Time: 15.5s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\frac{\sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \sqrt{3}}{2} + \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{1}{3} - \frac{2}{3} \cdot \pi\right) \cdot \frac{1}{2}\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\frac{\sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \sqrt{3}}{2} + \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{1}{3} - \frac{2}{3} \cdot \pi\right) \cdot \frac{1}{2}\right) \cdot 2
double f(double g, double h) {
        double r2370997 = 2.0;
        double r2370998 = atan2(1.0, 0.0);
        double r2370999 = r2370997 * r2370998;
        double r2371000 = 3.0;
        double r2371001 = r2370999 / r2371000;
        double r2371002 = g;
        double r2371003 = -r2371002;
        double r2371004 = h;
        double r2371005 = r2371003 / r2371004;
        double r2371006 = acos(r2371005);
        double r2371007 = r2371006 / r2371000;
        double r2371008 = r2371001 + r2371007;
        double r2371009 = cos(r2371008);
        double r2371010 = r2370997 * r2371009;
        return r2371010;
}
↓
double f(double g, double h) {
        double r2371011 = g;
        double r2371012 = h;
        double r2371013 = r2371011 / r2371012;
        double r2371014 = acos(r2371013);
        double r2371015 = 3.0;
        double r2371016 = r2371014 / r2371015;
        double r2371017 = atan2(1.0, 0.0);
        double r2371018 = 1.5;
        double r2371019 = r2371017 / r2371018;
        double r2371020 = r2371016 - r2371019;
        double r2371021 = sin(r2371020);
        double r2371022 = sqrt(r2371015);
        double r2371023 = r2371021 * r2371022;
        double r2371024 = 2.0;
        double r2371025 = r2371023 / r2371024;
        double r2371026 = 0.3333333333333333;
        double r2371027 = r2371014 * r2371026;
        double r2371028 = 0.6666666666666666;
        double r2371029 = r2371028 * r2371017;
        double r2371030 = r2371027 - r2371029;
        double r2371031 = cos(r2371030);
        double r2371032 = 0.5;
        double r2371033 = r2371031 * r2371032;
        double r2371034 = r2371025 + r2371033;
        double r2371035 = r2371034 * r2371024;
        return r2371035;
}

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

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

  4. Applied distribute-frac-neg1.0

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

  5. Applied acos-neg1.0

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

  6. Applied div-sub1.0

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

  7. Applied associate-+l-1.0

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

  8. Applied cos-diff0.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  11. Taylor expanded around 0 0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019155 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))