2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 15.6s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r1880414 = 2.0;
        double r1880415 = atan2(1.0, 0.0);
        double r1880416 = r1880414 * r1880415;
        double r1880417 = 3.0;
        double r1880418 = r1880416 / r1880417;
        double r1880419 = g;
        double r1880420 = -r1880419;
        double r1880421 = h;
        double r1880422 = r1880420 / r1880421;
        double r1880423 = acos(r1880422);
        double r1880424 = r1880423 / r1880417;
        double r1880425 = r1880418 + r1880424;
        double r1880426 = cos(r1880425);
        double r1880427 = r1880414 * r1880426;
        return r1880427;
}

↓

double f(double g, double h) {
        double r1880428 = 2.0;
        double r1880429 = 0.6666666666666666;
        double r1880430 = atan2(1.0, 0.0);
        double r1880431 = g;
        double r1880432 = -r1880431;
        double r1880433 = h;
        double r1880434 = r1880432 / r1880433;
        double r1880435 = acos(r1880434);
        double r1880436 = 3.0;
        double r1880437 = r1880435 / r1880436;
        double r1880438 = fma(r1880429, r1880430, r1880437);
        double r1880439 = cos(r1880438);
        double r1880440 = cbrt(r1880439);
        double r1880441 = r1880439 * r1880439;
        double r1880442 = cbrt(r1880441);
        double r1880443 = r1880440 * r1880442;
        double r1880444 = r1880428 * r1880443;
        return r1880444;
}

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

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

4. Applied add-cube-cbrt 1.0

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

6. Applied cbrt-unprod 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))