2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 5.3s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r155510 = 2.0;
        double r155511 = atan2(1.0, 0.0);
        double r155512 = r155510 * r155511;
        double r155513 = 3.0;
        double r155514 = r155512 / r155513;
        double r155515 = g;
        double r155516 = -r155515;
        double r155517 = h;
        double r155518 = r155516 / r155517;
        double r155519 = acos(r155518);
        double r155520 = r155519 / r155513;
        double r155521 = r155514 + r155520;
        double r155522 = cos(r155521);
        double r155523 = r155510 * r155522;
        return r155523;
}

↓

double f(double g, double h) {
        double r155524 = 2.0;
        double r155525 = 3.0;
        double r155526 = r155524 / r155525;
        double r155527 = atan2(1.0, 0.0);
        double r155528 = r155527 / r155525;
        double r155529 = fma(r155526, r155527, r155528);
        double r155530 = g;
        double r155531 = h;
        double r155532 = r155530 / r155531;
        double r155533 = acos(r155532);
        double r155534 = r155533 / r155525;
        double r155535 = r155529 - r155534;
        double r155536 = cos(r155535);
        double r155537 = r155536 * r155536;
        double r155538 = cbrt(r155537);
        double r155539 = cbrt(r155536);
        double r155540 = r155538 * r155539;
        double r155541 = r155524 * r155540;
        return r155541;
}

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 distribute-frac-neg 1.0

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

4. Applied acos-neg 1.0

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

5. Applied div-sub 1.0

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

6. Applied associate-+r- 1.0

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

7. Simplified 1.0

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

9. Applied add-cbrt-cube 1.0

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

10. Simplified 1.0

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

12. Applied add-cube-cbrt 1.0

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

13. Applied unpow-prod-down 1.0

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

14. Applied cbrt-prod 0.1

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

15. Simplified 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))