Average Error: 1.0 → 0.0
Time: 9.7s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right) \cdot \sqrt[3]{\pi}\right)\right)\]
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{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right) \cdot \sqrt[3]{\pi}\right)\right)
double f(double g, double h) {
        double r148171 = 2.0;
        double r148172 = atan2(1.0, 0.0);
        double r148173 = r148171 * r148172;
        double r148174 = 3.0;
        double r148175 = r148173 / r148174;
        double r148176 = g;
        double r148177 = -r148176;
        double r148178 = h;
        double r148179 = r148177 / r148178;
        double r148180 = acos(r148179);
        double r148181 = r148180 / r148174;
        double r148182 = r148175 + r148181;
        double r148183 = cos(r148182);
        double r148184 = r148171 * r148183;
        return r148184;
}
↓
double f(double g, double h) {
        double r148185 = 2.0;
        double r148186 = g;
        double r148187 = -r148186;
        double r148188 = h;
        double r148189 = r148187 / r148188;
        double r148190 = acos(r148189);
        double r148191 = 3.0;
        double r148192 = r148190 / r148191;
        double r148193 = cos(r148192);
        double r148194 = r148185 / r148191;
        double r148195 = atan2(1.0, 0.0);
        double r148196 = r148194 * r148195;
        double r148197 = cos(r148196);
        double r148198 = r148193 * r148197;
        double r148199 = sin(r148192);
        double r148200 = cbrt(r148195);
        double r148201 = r148200 * r148200;
        double r148202 = r148194 * r148201;
        double r148203 = r148202 * r148200;
        double r148204 = sin(r148203);
        double r148205 = r148199 * r148204;
        double r148206 = r148198 - r148205;
        double r148207 = r148185 * r148206;
        return r148207;
}

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-sqr-sqrt1.0

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

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

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

  6. Applied times-frac1.0

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

  8. Applied fma-udef1.0

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

  9. Applied cos-sum1.0

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

  10. Simplified0.0

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

  11. Simplified1.0

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

  13. Applied add-cube-cbrt0.0

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

  14. Applied associate-*r*0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))