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

Average Error: 13.0 → 0.2

Time: 9.9s

Precision: 64

\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r165877 = x;
        double r165878 = y;
        double r165879 = z;
        double r165880 = r165878 + r165879;
        double r165881 = tan(r165880);
        double r165882 = a;
        double r165883 = tan(r165882);
        double r165884 = r165881 - r165883;
        double r165885 = r165877 + r165884;
        return r165885;
}

↓

double f(double x, double y, double z, double a) {
        double r165886 = x;
        double r165887 = y;
        double r165888 = tan(r165887);
        double r165889 = z;
        double r165890 = tan(r165889);
        double r165891 = r165888 + r165890;
        double r165892 = a;
        double r165893 = cos(r165892);
        double r165894 = r165891 * r165893;
        double r165895 = 1.0;
        double r165896 = r165888 * r165890;
        double r165897 = r165895 - r165896;
        double r165898 = sin(r165892);
        double r165899 = r165897 * r165898;
        double r165900 = r165894 - r165899;
        double r165901 = 3.0;
        double r165902 = pow(r165895, r165901);
        double r165903 = pow(r165896, r165901);
        double r165904 = r165902 - r165903;
        double r165905 = r165904 * r165893;
        double r165906 = r165900 / r165905;
        double r165907 = r165895 * r165895;
        double r165908 = r165896 * r165896;
        double r165909 = cbrt(r165888);
        double r165910 = r165909 * r165909;
        double r165911 = r165909 * r165890;
        double r165912 = r165910 * r165911;
        double r165913 = r165895 * r165912;
        double r165914 = r165908 + r165913;
        double r165915 = r165907 + r165914;
        double r165916 = r165906 * r165915;
        double r165917 = r165886 + r165916;
        return r165917;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.0

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
2. Using strategy rm

3. Applied tan-quot 13.0

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

4. Applied tan-sum 0.2

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

5. Applied frac-sub 0.2

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

7. Applied flip3-- 0.2

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

8. Applied associate-*l/ 0.2

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

9. Applied associate-/r/ 0.2

   \[\leadsto x + \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left({1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}\right) \cdot \cos a} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)}\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.2

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

12. Applied associate-*l* 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :precision binary64
  :pre (and (or (== x 0.0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.7512240000000001e+308)) (or (<= -1.7767070000000002e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.7512240000000001e+308)))
  (+ x (- (tan (+ y z)) (tan a))))