Average Error: 1.0 → 0.0
Time: 21.9s
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 r115252 = 2.0;
        double r115253 = atan2(1.0, 0.0);
        double r115254 = r115252 * r115253;
        double r115255 = 3.0;
        double r115256 = r115254 / r115255;
        double r115257 = g;
        double r115258 = -r115257;
        double r115259 = h;
        double r115260 = r115258 / r115259;
        double r115261 = acos(r115260);
        double r115262 = r115261 / r115255;
        double r115263 = r115256 + r115262;
        double r115264 = cos(r115263);
        double r115265 = r115252 * r115264;
        return r115265;
}
↓
double f(double g, double h) {
        double r115266 = 2.0;
        double r115267 = 3.0;
        double r115268 = r115266 / r115267;
        double r115269 = atan2(1.0, 0.0);
        double r115270 = g;
        double r115271 = -r115270;
        double r115272 = h;
        double r115273 = r115271 / r115272;
        double r115274 = acos(r115273);
        double r115275 = r115274 / r115267;
        double r115276 = fma(r115268, r115269, r115275);
        double r115277 = cos(r115276);
        double r115278 = 2.0;
        double r115279 = pow(r115277, r115278);
        double r115280 = cbrt(r115279);
        double r115281 = cbrt(r115277);
        double r115282 = r115280 * r115281;
        double r115283 = 3.0;
        double r115284 = pow(r115282, r115283);
        double r115285 = cbrt(r115284);
        double r115286 = r115266 * r115285;
        return r115286;
}

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. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))