2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 35.9s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r5703605 = 2.0;
        double r5703606 = atan2(1.0, 0.0);
        double r5703607 = r5703605 * r5703606;
        double r5703608 = 3.0;
        double r5703609 = r5703607 / r5703608;
        double r5703610 = g;
        double r5703611 = -r5703610;
        double r5703612 = h;
        double r5703613 = r5703611 / r5703612;
        double r5703614 = acos(r5703613);
        double r5703615 = r5703614 / r5703608;
        double r5703616 = r5703609 + r5703615;
        double r5703617 = cos(r5703616);
        double r5703618 = r5703605 * r5703617;
        return r5703618;
}

↓

double f(double g, double h) {
        double r5703619 = 2.0;
        double r5703620 = g;
        double r5703621 = h;
        double r5703622 = r5703620 / r5703621;
        double r5703623 = -r5703622;
        double r5703624 = acos(r5703623);
        double r5703625 = 3.0;
        double r5703626 = sqrt(r5703625);
        double r5703627 = r5703624 / r5703626;
        double r5703628 = r5703627 / r5703626;
        double r5703629 = sqrt(r5703628);
        double r5703630 = r5703629 * r5703629;
        double r5703631 = cos(r5703630);
        double r5703632 = 0.6666666666666666;
        double r5703633 = atan2(1.0, 0.0);
        double r5703634 = r5703632 * r5703633;
        double r5703635 = cos(r5703634);
        double r5703636 = r5703631 * r5703635;
        double r5703637 = sin(r5703634);
        double r5703638 = sqrt(r5703637);
        double r5703639 = r5703638 * r5703638;
        double r5703640 = sin(r5703630);
        double r5703641 = r5703639 * r5703640;
        double r5703642 = r5703636 - r5703641;
        double r5703643 = r5703619 * r5703642;
        return r5703643;
}

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-sqr-sqrt 1.0

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

5. Applied associate-/r* 1.0

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

7. Applied add-sqr-sqrt 1.0

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

9. Applied fma-udef 1.0

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

10. Applied cos-sum 0.0

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

12. Applied add-sqr-sqrt 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))