2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 15.8s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r24372 = 2.0;
        double r24373 = atan2(1.0, 0.0);
        double r24374 = r24372 * r24373;
        double r24375 = 3.0;
        double r24376 = r24374 / r24375;
        double r24377 = g;
        double r24378 = -r24377;
        double r24379 = h;
        double r24380 = r24378 / r24379;
        double r24381 = acos(r24380);
        double r24382 = r24381 / r24375;
        double r24383 = r24376 + r24382;
        double r24384 = cos(r24383);
        double r24385 = r24372 * r24384;
        return r24385;
}

↓

double f(double g, double h) {
        double r24386 = 2.0;
        double r24387 = atan2(1.0, 0.0);
        double r24388 = r24386 * r24387;
        double r24389 = 3.0;
        double r24390 = r24388 / r24389;
        double r24391 = g;
        double r24392 = -r24391;
        double r24393 = h;
        double r24394 = r24392 / r24393;
        double r24395 = acos(r24394);
        double r24396 = r24395 / r24389;
        double r24397 = r24390 + r24396;
        double r24398 = cos(r24397);
        double r24399 = 2.0;
        double r24400 = pow(r24398, r24399);
        double r24401 = cbrt(r24400);
        double r24402 = cbrt(r24398);
        double r24403 = r24401 * r24402;
        double r24404 = r24386 * r24403;
        return r24404;
}

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 add-cube-cbrt 1.0

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

7. Applied unpow-prod-down 1.0

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

8. Applied cbrt-prod 0.1

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

9. Simplified 0.0

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

10. Simplified 0.0

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

11. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019310 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))