(+ x (- (tan (+ y z)) (tan a)))

Average Error: 13.1 → 0.2

Time: 33.1s

Precision: 64

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]

Math

\[x + \left(\tan \left(y + z\right) - \tan a\right)\]

↓

\[x - \left(\left(\tan a - \frac{\left(\tan y + \tan z\right) \cdot \mathsf{fma}\left(\tan z, \tan y, 1\right)}{1 - \tan z \cdot \left(\sqrt[3]{{\left(\tan y \cdot \tan y\right)}^{3}} \cdot \tan z\right)}\right) + \left(\mathsf{fma}\left(\tan z, \tan y, 1\right) + \left(-1 - \tan y \cdot \tan z\right)\right) \cdot \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)}\right)\]

C

double f(double x, double y, double z, double a) {
        double r121770 = x;
        double r121771 = y;
        double r121772 = z;
        double r121773 = r121771 + r121772;
        double r121774 = tan(r121773);
        double r121775 = a;
        double r121776 = tan(r121775);
        double r121777 = r121774 - r121776;
        double r121778 = r121770 + r121777;
        return r121778;
}

↓

double f(double x, double y, double z, double a) {
        double r121779 = x;
        double r121780 = a;
        double r121781 = tan(r121780);
        double r121782 = y;
        double r121783 = tan(r121782);
        double r121784 = z;
        double r121785 = tan(r121784);
        double r121786 = r121783 + r121785;
        double r121787 = 1.0;
        double r121788 = fma(r121785, r121783, r121787);
        double r121789 = r121786 * r121788;
        double r121790 = r121783 * r121783;
        double r121791 = 3.0;
        double r121792 = pow(r121790, r121791);
        double r121793 = cbrt(r121792);
        double r121794 = r121793 * r121785;
        double r121795 = r121785 * r121794;
        double r121796 = r121787 - r121795;
        double r121797 = r121789 / r121796;
        double r121798 = r121781 - r121797;
        double r121799 = -1.0;
        double r121800 = r121783 * r121785;
        double r121801 = r121799 - r121800;
        double r121802 = r121788 + r121801;
        double r121803 = r121790 * r121785;
        double r121804 = r121785 * r121803;
        double r121805 = r121787 - r121804;
        double r121806 = r121786 / r121805;
        double r121807 = r121802 * r121806;
        double r121808 = r121798 + r121807;
        double r121809 = r121779 - r121808;
        return r121809;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.1

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]

2. Simplified 13.1

   \[\leadsto \color{blue}{x - \left(\tan a - \tan \left(y + z\right)\right)}\]
3. Using strategy rm

4. Applied tan-sum 0.2

   \[\leadsto x - \left(\tan a - \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}}\right)\]

5. Simplified 0.2

   \[\leadsto x - \left(\tan a - \frac{\color{blue}{\tan z + \tan y}}{1 - \tan y \cdot \tan z}\right)\]

6. Simplified 0.2

   \[\leadsto x - \left(\tan a - \frac{\tan z + \tan y}{\color{blue}{1 - \tan z \cdot \tan y}}\right)\]
7. Using strategy rm

8. Applied flip-- 0.2

   \[\leadsto x - \left(\tan a - \frac{\tan z + \tan y}{\color{blue}{\frac{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}{1 + \tan z \cdot \tan y}}}\right)\]

9. Applied associate-/r/ 0.2

   \[\leadsto x - \left(\tan a - \color{blue}{\frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)} \cdot \left(1 + \tan z \cdot \tan y\right)}\right)\]

10. Applied add-sqr-sqrt 32.5

    \[\leadsto x - \left(\color{blue}{\sqrt{\tan a} \cdot \sqrt{\tan a}} - \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)} \cdot \left(1 + \tan z \cdot \tan y\right)\right)\]

11. Applied prod-diff 32.5

    \[\leadsto x - \color{blue}{\left(\mathsf{fma}\left(\sqrt{\tan a}, \sqrt{\tan a}, -\left(1 + \tan z \cdot \tan y\right) \cdot \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) + \mathsf{fma}\left(-\left(1 + \tan z \cdot \tan y\right), \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}, \left(1 + \tan z \cdot \tan y\right) \cdot \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right)\right)}\]

12. Simplified 0.2

    \[\leadsto x - \left(\color{blue}{\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)}\right)} + \mathsf{fma}\left(-\left(1 + \tan z \cdot \tan y\right), \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}, \left(1 + \tan z \cdot \tan y\right) \cdot \frac{\tan z + \tan y}{1 \cdot 1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right)\right)\]

13. Simplified 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)}\right) + \color{blue}{\frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)} \cdot \left(\left(-1 - \tan y \cdot \tan z\right) + \mathsf{fma}\left(\tan z, \tan y, 1\right)\right)}\right)\]
14. Using strategy rm

15. Applied add-cbrt-cube 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\left(\tan y \cdot \color{blue}{\sqrt[3]{\left(\tan y \cdot \tan y\right) \cdot \tan y}}\right) \cdot \tan z\right)}\right) + \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)} \cdot \left(\left(-1 - \tan y \cdot \tan z\right) + \mathsf{fma}\left(\tan z, \tan y, 1\right)\right)\right)\]

16. Applied add-cbrt-cube 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\left(\color{blue}{\sqrt[3]{\left(\tan y \cdot \tan y\right) \cdot \tan y}} \cdot \sqrt[3]{\left(\tan y \cdot \tan y\right) \cdot \tan y}\right) \cdot \tan z\right)}\right) + \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)} \cdot \left(\left(-1 - \tan y \cdot \tan z\right) + \mathsf{fma}\left(\tan z, \tan y, 1\right)\right)\right)\]

17. Applied cbrt-unprod 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right) \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan y\right)}} \cdot \tan z\right)}\right) + \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)} \cdot \left(\left(-1 - \tan y \cdot \tan z\right) + \mathsf{fma}\left(\tan z, \tan y, 1\right)\right)\right)\]

18. Simplified 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\mathsf{fma}\left(\tan z, \tan y, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - \tan z \cdot \left(\sqrt[3]{\color{blue}{{\left(\tan y \cdot \tan y\right)}^{3}}} \cdot \tan z\right)}\right) + \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)} \cdot \left(\left(-1 - \tan y \cdot \tan z\right) + \mathsf{fma}\left(\tan z, \tan y, 1\right)\right)\right)\]

19. Final simplification 0.2

    \[\leadsto x - \left(\left(\tan a - \frac{\left(\tan y + \tan z\right) \cdot \mathsf{fma}\left(\tan z, \tan y, 1\right)}{1 - \tan z \cdot \left(\sqrt[3]{{\left(\tan y \cdot \tan y\right)}^{3}} \cdot \tan z\right)}\right) + \left(\mathsf{fma}\left(\tan z, \tan y, 1\right) + \left(-1 - \tan y \cdot \tan z\right)\right) \cdot \frac{\tan y + \tan z}{1 - \tan z \cdot \left(\left(\tan y \cdot \tan y\right) \cdot \tan z\right)}\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0.0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))