2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 44.4s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r9383067 = 2.0;
        double r9383068 = atan2(1.0, 0.0);
        double r9383069 = r9383067 * r9383068;
        double r9383070 = 3.0;
        double r9383071 = r9383069 / r9383070;
        double r9383072 = g;
        double r9383073 = -r9383072;
        double r9383074 = h;
        double r9383075 = r9383073 / r9383074;
        double r9383076 = acos(r9383075);
        double r9383077 = r9383076 / r9383070;
        double r9383078 = r9383071 + r9383077;
        double r9383079 = cos(r9383078);
        double r9383080 = r9383067 * r9383079;
        return r9383080;
}

↓

double f(double g, double h) {
        double r9383081 = 0.6666666666666666;
        double r9383082 = atan2(1.0, 0.0);
        double r9383083 = g;
        double r9383084 = -r9383083;
        double r9383085 = h;
        double r9383086 = r9383084 / r9383085;
        double r9383087 = acos(r9383086);
        double r9383088 = 3.0;
        double r9383089 = r9383087 / r9383088;
        double r9383090 = fma(r9383081, r9383082, r9383089);
        double r9383091 = cos(r9383090);
        double r9383092 = exp(r9383091);
        double r9383093 = cbrt(r9383092);
        double r9383094 = r9383093 * r9383093;
        double r9383095 = r9383094 * r9383093;
        double r9383096 = log(r9383095);
        double r9383097 = 2.0;
        double r9383098 = r9383096 * r9383097;
        return r9383098;
}

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. Simplified 1.0

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

4. Applied add-log-exp 1.0

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

6. Applied add-cube-cbrt 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))