2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 3.9s

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 r111152 = 2.0;
        double r111153 = atan2(1.0, 0.0);
        double r111154 = r111152 * r111153;
        double r111155 = 3.0;
        double r111156 = r111154 / r111155;
        double r111157 = g;
        double r111158 = -r111157;
        double r111159 = h;
        double r111160 = r111158 / r111159;
        double r111161 = acos(r111160);
        double r111162 = r111161 / r111155;
        double r111163 = r111156 + r111162;
        double r111164 = cos(r111163);
        double r111165 = r111152 * r111164;
        return r111165;
}

↓

double f(double g, double h) {
        double r111166 = 2.0;
        double r111167 = atan2(1.0, 0.0);
        double r111168 = r111166 * r111167;
        double r111169 = 3.0;
        double r111170 = r111168 / r111169;
        double r111171 = g;
        double r111172 = -r111171;
        double r111173 = h;
        double r111174 = r111172 / r111173;
        double r111175 = acos(r111174);
        double r111176 = r111175 / r111169;
        double r111177 = r111170 + r111176;
        double r111178 = cos(r111177);
        double r111179 = cbrt(r111178);
        double r111180 = r111178 * r111178;
        double r111181 = cbrt(r111180);
        double r111182 = r111179 * r111181;
        double r111183 = r111166 * r111182;
        return r111183;
}

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 2020047 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))