Average Error: 1.0 → 0.0
Time: 23.4s
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 r107061 = 2.0;
        double r107062 = atan2(1.0, 0.0);
        double r107063 = r107061 * r107062;
        double r107064 = 3.0;
        double r107065 = r107063 / r107064;
        double r107066 = g;
        double r107067 = -r107066;
        double r107068 = h;
        double r107069 = r107067 / r107068;
        double r107070 = acos(r107069);
        double r107071 = r107070 / r107064;
        double r107072 = r107065 + r107071;
        double r107073 = cos(r107072);
        double r107074 = r107061 * r107073;
        return r107074;
}
↓
double f(double g, double h) {
        double r107075 = 2.0;
        double r107076 = 3.0;
        double r107077 = r107075 / r107076;
        double r107078 = atan2(1.0, 0.0);
        double r107079 = g;
        double r107080 = -r107079;
        double r107081 = h;
        double r107082 = r107080 / r107081;
        double r107083 = acos(r107082);
        double r107084 = r107083 / r107076;
        double r107085 = fma(r107077, r107078, r107084);
        double r107086 = cos(r107085);
        double r107087 = 2.0;
        double r107088 = pow(r107086, r107087);
        double r107089 = cbrt(r107088);
        double r107090 = cbrt(r107086);
        double r107091 = r107089 * r107090;
        double r107092 = 3.0;
        double r107093 = pow(r107091, r107092);
        double r107094 = cbrt(r107093);
        double r107095 = r107075 * r107094;
        return r107095;
}

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)))))