Average Error: 1.0 → 0.0
Time: 18.8s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot \frac{2}{3}\right) \cdot \sqrt{\pi}\right)\right)\]
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{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin \left(\frac{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot \frac{2}{3}\right) \cdot \sqrt{\pi}\right)\right)
double f(double g, double h) {
        double r103211 = 2.0;
        double r103212 = atan2(1.0, 0.0);
        double r103213 = r103211 * r103212;
        double r103214 = 3.0;
        double r103215 = r103213 / r103214;
        double r103216 = g;
        double r103217 = -r103216;
        double r103218 = h;
        double r103219 = r103217 / r103218;
        double r103220 = acos(r103219);
        double r103221 = r103220 / r103214;
        double r103222 = r103215 + r103221;
        double r103223 = cos(r103222);
        double r103224 = r103211 * r103223;
        return r103224;
}
↓
double f(double g, double h) {
        double r103225 = 2.0;
        double r103226 = 3.0;
        double r103227 = r103225 / r103226;
        double r103228 = atan2(1.0, 0.0);
        double r103229 = r103227 * r103228;
        double r103230 = cos(r103229);
        double r103231 = g;
        double r103232 = -r103231;
        double r103233 = h;
        double r103234 = r103232 / r103233;
        double r103235 = acos(r103234);
        double r103236 = cbrt(r103226);
        double r103237 = r103236 * r103236;
        double r103238 = r103235 / r103237;
        double r103239 = r103238 / r103236;
        double r103240 = cos(r103239);
        double r103241 = r103230 * r103240;
        double r103242 = sin(r103239);
        double r103243 = sqrt(r103228);
        double r103244 = r103243 * r103227;
        double r103245 = r103244 * r103243;
        double r103246 = sin(r103245);
        double r103247 = r103242 * r103246;
        double r103248 = r103241 - r103247;
        double r103249 = r103225 * r103248;
        return r103249;
}

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 add-cube-cbrt1.0

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

  5. Applied *-un-lft-identity1.0

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

  6. Applied times-frac1.0

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

  8. Applied fma-udef1.0

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

  9. Applied cos-sum1.0

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

  10. Simplified1.0

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

  11. Simplified1.0

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

  13. Applied add-sqr-sqrt0.0

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

  14. Applied associate-*r*0.0

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

  15. Simplified0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 +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)))))