Average Error: 1.0 → 0.0
Time: 30.8s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\sqrt{\pi} \cdot \left(\frac{2}{3} \cdot \sqrt{\pi}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\sqrt{\pi} \cdot \left(\frac{2}{3} \cdot \sqrt{\pi}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2
double f(double g, double h) {
        double r6617227 = 2.0;
        double r6617228 = atan2(1.0, 0.0);
        double r6617229 = r6617227 * r6617228;
        double r6617230 = 3.0;
        double r6617231 = r6617229 / r6617230;
        double r6617232 = g;
        double r6617233 = -r6617232;
        double r6617234 = h;
        double r6617235 = r6617233 / r6617234;
        double r6617236 = acos(r6617235);
        double r6617237 = r6617236 / r6617230;
        double r6617238 = r6617231 + r6617237;
        double r6617239 = cos(r6617238);
        double r6617240 = r6617227 * r6617239;
        return r6617240;
}
↓
double f(double g, double h) {
        double r6617241 = g;
        double r6617242 = -r6617241;
        double r6617243 = h;
        double r6617244 = r6617242 / r6617243;
        double r6617245 = acos(r6617244);
        double r6617246 = 3.0;
        double r6617247 = r6617245 / r6617246;
        double r6617248 = cos(r6617247);
        double r6617249 = 2.0;
        double r6617250 = r6617249 / r6617246;
        double r6617251 = atan2(1.0, 0.0);
        double r6617252 = r6617250 * r6617251;
        double r6617253 = cos(r6617252);
        double r6617254 = r6617248 * r6617253;
        double r6617255 = sqrt(r6617251);
        double r6617256 = r6617250 * r6617255;
        double r6617257 = r6617255 * r6617256;
        double r6617258 = sin(r6617257);
        double r6617259 = sin(r6617247);
        double r6617260 = r6617258 * r6617259;
        double r6617261 = r6617254 - r6617260;
        double r6617262 = r6617261 * r6617249;
        return r6617262;
}

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

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

  4. Applied fma-udef1.0

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

  5. Applied cos-sum1.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied associate-*r*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))