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

Average Error: 12.8 → 0.2

Time: 14.2s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r182836 = x;
        double r182837 = y;
        double r182838 = z;
        double r182839 = r182837 + r182838;
        double r182840 = tan(r182839);
        double r182841 = a;
        double r182842 = tan(r182841);
        double r182843 = r182840 - r182842;
        double r182844 = r182836 + r182843;
        return r182844;
}

↓

double f(double x, double y, double z, double a) {
        double r182845 = x;
        double r182846 = 1.0;
        double r182847 = y;
        double r182848 = sin(r182847);
        double r182849 = z;
        double r182850 = sin(r182849);
        double r182851 = r182848 * r182850;
        double r182852 = cos(r182849);
        double r182853 = cos(r182847);
        double r182854 = r182852 * r182853;
        double r182855 = r182851 / r182854;
        double r182856 = r182846 - r182855;
        double r182857 = r182846 / r182856;
        double r182858 = r182848 / r182853;
        double r182859 = r182850 / r182852;
        double r182860 = r182858 + r182859;
        double r182861 = r182857 * r182860;
        double r182862 = a;
        double r182863 = sin(r182862);
        double r182864 = cos(r182862);
        double r182865 = r182863 / r182864;
        double r182866 = r182861 - r182865;
        double r182867 = -r182846;
        double r182868 = r182867 + r182846;
        double r182869 = r182865 * r182868;
        double r182870 = r182866 + r182869;
        double r182871 = r182845 + r182870;
        return r182871;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.8

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

3. Applied tan-sum 0.2

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

4. Taylor expanded around inf 0.2

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

6. Applied add-cube-cbrt 0.3

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

7. Applied *-un-lft-identity 0.3

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

8. Applied times-frac 0.3

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

9. Applied *-un-lft-identity 0.3

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

10. Applied times-frac 0.3

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

11. Applied distribute-lft-out 0.3

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

12. Applied prod-diff 0.3

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

13. Simplified 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020036 +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))))