Average Error: 1.0 → 0.1
Time: 22.0s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}^{2}} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)}^{3}}\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)}^{2}} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)}^{3}}
double f(double g, double h) {
        double r83251 = 2.0;
        double r83252 = atan2(1.0, 0.0);
        double r83253 = r83251 * r83252;
        double r83254 = 3.0;
        double r83255 = r83253 / r83254;
        double r83256 = g;
        double r83257 = -r83256;
        double r83258 = h;
        double r83259 = r83257 / r83258;
        double r83260 = acos(r83259);
        double r83261 = r83260 / r83254;
        double r83262 = r83255 + r83261;
        double r83263 = cos(r83262);
        double r83264 = r83251 * r83263;
        return r83264;
}
↓
double f(double g, double h) {
        double r83265 = 2.0;
        double r83266 = 3.0;
        double r83267 = r83265 / r83266;
        double r83268 = atan2(1.0, 0.0);
        double r83269 = g;
        double r83270 = -r83269;
        double r83271 = h;
        double r83272 = r83270 / r83271;
        double r83273 = acos(r83272);
        double r83274 = r83273 / r83266;
        double r83275 = fma(r83267, r83268, r83274);
        double r83276 = cos(r83275);
        double r83277 = 2.0;
        double r83278 = pow(r83276, r83277);
        double r83279 = cbrt(r83278);
        double r83280 = cbrt(r83276);
        double r83281 = r83279 * r83280;
        double r83282 = 3.0;
        double r83283 = pow(r83281, r83282);
        double r83284 = cbrt(r83283);
        double r83285 = r83265 * r83284;
        return r83285;
}

Error

Bits error versus g

Bits error versus h

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-cbrt-cube1.5

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

  5. Simplified1.0

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

  7. Applied add-cube-cbrt1.0

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))