Average Error: 1.0 → 0.0
Time: 22.2s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right)\right) \cdot 2\]
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{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(-\frac{g}{h}\right)}}}\right)\right) \cdot 2
double f(double g, double h) {
        double r4394215 = 2.0;
        double r4394216 = atan2(1.0, 0.0);
        double r4394217 = r4394215 * r4394216;
        double r4394218 = 3.0;
        double r4394219 = r4394217 / r4394218;
        double r4394220 = g;
        double r4394221 = -r4394220;
        double r4394222 = h;
        double r4394223 = r4394221 / r4394222;
        double r4394224 = acos(r4394223);
        double r4394225 = r4394224 / r4394218;
        double r4394226 = r4394219 + r4394225;
        double r4394227 = cos(r4394226);
        double r4394228 = r4394215 * r4394227;
        return r4394228;
}
↓
double f(double g, double h) {
        double r4394229 = 0.6666666666666666;
        double r4394230 = atan2(1.0, 0.0);
        double r4394231 = r4394229 * r4394230;
        double r4394232 = cos(r4394231);
        double r4394233 = g;
        double r4394234 = h;
        double r4394235 = r4394233 / r4394234;
        double r4394236 = -r4394235;
        double r4394237 = acos(r4394236);
        double r4394238 = sqrt(r4394237);
        double r4394239 = 3.0;
        double r4394240 = r4394239 / r4394238;
        double r4394241 = r4394238 / r4394240;
        double r4394242 = cos(r4394241);
        double r4394243 = r4394232 * r4394242;
        double r4394244 = sqrt(r4394230);
        double r4394245 = r4394244 * r4394229;
        double r4394246 = r4394244 * r4394245;
        double r4394247 = sin(r4394246);
        double r4394248 = sin(r4394241);
        double r4394249 = r4394247 * r4394248;
        double r4394250 = r4394243 - r4394249;
        double r4394251 = 2.0;
        double r4394252 = r4394250 * r4394251;
        return r4394252;
}

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

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

  5. Applied associate-/l*1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied associate-*r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))