Average Error: 1.0 → 0.0
Time: 18.4s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)\]
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) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)
double f(double g, double h) {
        double r6497256 = 2.0;
        double r6497257 = atan2(1.0, 0.0);
        double r6497258 = r6497256 * r6497257;
        double r6497259 = 3.0;
        double r6497260 = r6497258 / r6497259;
        double r6497261 = g;
        double r6497262 = -r6497261;
        double r6497263 = h;
        double r6497264 = r6497262 / r6497263;
        double r6497265 = acos(r6497264);
        double r6497266 = r6497265 / r6497259;
        double r6497267 = r6497260 + r6497266;
        double r6497268 = cos(r6497267);
        double r6497269 = r6497256 * r6497268;
        return r6497269;
}
↓
double f(double g, double h) {
        double r6497270 = 2.0;
        double r6497271 = g;
        double r6497272 = h;
        double r6497273 = r6497271 / r6497272;
        double r6497274 = acos(r6497273);
        double r6497275 = 1.5;
        double r6497276 = r6497274 * r6497275;
        double r6497277 = 3.0;
        double r6497278 = atan2(1.0, 0.0);
        double r6497279 = r6497277 * r6497278;
        double r6497280 = r6497276 - r6497279;
        double r6497281 = 4.5;
        double r6497282 = r6497280 / r6497281;
        double r6497283 = cos(r6497282);
        double r6497284 = 0.5;
        double r6497285 = r6497283 * r6497284;
        double r6497286 = sqrt(r6497277);
        double r6497287 = r6497286 / r6497270;
        double r6497288 = r6497274 / r6497277;
        double r6497289 = r6497278 / r6497275;
        double r6497290 = r6497288 - r6497289;
        double r6497291 = sin(r6497290);
        double r6497292 = r6497287 * r6497291;
        double r6497293 = r6497285 + r6497292;
        double r6497294 = r6497270 * r6497293;
        return r6497294;
}

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

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

  12. Applied frac-sub0.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right) \cdot \frac{3}{2} - 3 \cdot \pi}{\frac{9}{2}}\right) \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))