2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 16.4s

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 r3051158 = 2.0;
        double r3051159 = atan2(1.0, 0.0);
        double r3051160 = r3051158 * r3051159;
        double r3051161 = 3.0;
        double r3051162 = r3051160 / r3051161;
        double r3051163 = g;
        double r3051164 = -r3051163;
        double r3051165 = h;
        double r3051166 = r3051164 / r3051165;
        double r3051167 = acos(r3051166);
        double r3051168 = r3051167 / r3051161;
        double r3051169 = r3051162 + r3051168;
        double r3051170 = cos(r3051169);
        double r3051171 = r3051158 * r3051170;
        return r3051171;
}

↓

double f(double g, double h) {
        double r3051172 = 0.6666666666666666;
        double r3051173 = atan2(1.0, 0.0);
        double r3051174 = g;
        double r3051175 = -r3051174;
        double r3051176 = h;
        double r3051177 = r3051175 / r3051176;
        double r3051178 = acos(r3051177);
        double r3051179 = 3.0;
        double r3051180 = r3051178 / r3051179;
        double r3051181 = fma(r3051172, r3051173, r3051180);
        double r3051182 = cos(r3051181);
        double r3051183 = exp(r3051182);
        double r3051184 = cbrt(r3051183);
        double r3051185 = r3051184 * r3051184;
        double r3051186 = r3051185 * r3051184;
        double r3051187 = log(r3051186);
        double r3051188 = 2.0;
        double r3051189 = r3051187 * r3051188;
        return r3051189;
}

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)))))