2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 13.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double g, double h) {
        double r165319 = 2.0;
        double r165320 = atan2(1.0, 0.0);
        double r165321 = r165319 * r165320;
        double r165322 = 3.0;
        double r165323 = r165321 / r165322;
        double r165324 = g;
        double r165325 = -r165324;
        double r165326 = h;
        double r165327 = r165325 / r165326;
        double r165328 = acos(r165327);
        double r165329 = r165328 / r165322;
        double r165330 = r165323 + r165329;
        double r165331 = cos(r165330);
        double r165332 = r165319 * r165331;
        return r165332;
}

↓

double f(double g, double h) {
        double r165333 = 2.0;
        double r165334 = atan2(1.0, 0.0);
        double r165335 = 3.0;
        double r165336 = r165333 / r165335;
        double r165337 = g;
        double r165338 = h;
        double r165339 = r165337 / r165338;
        double r165340 = -r165339;
        double r165341 = acos(r165340);
        double r165342 = r165341 / r165335;
        double r165343 = fma(r165334, r165336, r165342);
        double r165344 = cos(r165343);
        double r165345 = cbrt(r165344);
        double r165346 = 2.0;
        double r165347 = pow(r165344, r165346);
        double r165348 = cbrt(r165347);
        double r165349 = r165345 * r165348;
        double r165350 = r165333 * r165349;
        return r165350;
}

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

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

4. Applied add-cube-cbrt 1.0

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

5. Applied add-cube-cbrt 1.0

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

6. Applied times-frac 1.0

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

8. Applied add-cbrt-cube 1.5

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

9. Simplified 1.0

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

11. Applied cube-mult 1.0

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

12. Applied cbrt-prod 0.0

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

13. Simplified 0.0

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

14. Simplified 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))