2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 5.5s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r159940 = 2.0;
        double r159941 = atan2(1.0, 0.0);
        double r159942 = r159940 * r159941;
        double r159943 = 3.0;
        double r159944 = r159942 / r159943;
        double r159945 = g;
        double r159946 = -r159945;
        double r159947 = h;
        double r159948 = r159946 / r159947;
        double r159949 = acos(r159948);
        double r159950 = r159949 / r159943;
        double r159951 = r159944 + r159950;
        double r159952 = cos(r159951);
        double r159953 = r159940 * r159952;
        return r159953;
}

↓

double f(double g, double h) {
        double r159954 = 2.0;
        double r159955 = atan2(1.0, 0.0);
        double r159956 = r159954 * r159955;
        double r159957 = 3.0;
        double r159958 = r159956 / r159957;
        double r159959 = g;
        double r159960 = -r159959;
        double r159961 = h;
        double r159962 = r159960 / r159961;
        double r159963 = acos(r159962);
        double r159964 = r159963 / r159957;
        double r159965 = r159958 + r159964;
        double r159966 = cos(r159965);
        double r159967 = cbrt(r159966);
        double r159968 = r159966 * r159966;
        double r159969 = cbrt(r159968);
        double r159970 = r159967 * r159969;
        double r159971 = r159954 * r159970;
        return r159971;
}

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. Using strategy rm

3. Applied add-cbrt-cube 1.6

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

4. Simplified 1.0

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

6. Applied cube-mult 1.6

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

7. Applied cbrt-prod 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019318 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))