Average Error: 13.1 → 0.3
Time: 19.5s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(x + \left(\frac{\sin y}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos z}\right)\right) - \frac{\sin a}{\cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left(x + \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)\right) - \frac{\sin a}{\cos a}
double f(double x, double y, double z, double a) {
        double r124827 = x;
        double r124828 = y;
        double r124829 = z;
        double r124830 = r124828 + r124829;
        double r124831 = tan(r124830);
        double r124832 = a;
        double r124833 = tan(r124832);
        double r124834 = r124831 - r124833;
        double r124835 = r124827 + r124834;
        return r124835;
}
↓
double f(double x, double y, double z, double a) {
        double r124836 = x;
        double r124837 = y;
        double r124838 = sin(r124837);
        double r124839 = 1.0;
        double r124840 = z;
        double r124841 = sin(r124840);
        double r124842 = r124838 * r124841;
        double r124843 = cos(r124840);
        double r124844 = cos(r124837);
        double r124845 = r124843 * r124844;
        double r124846 = r124842 / r124845;
        double r124847 = r124839 - r124846;
        double r124848 = r124847 * r124844;
        double r124849 = r124838 / r124848;
        double r124850 = r124847 * r124843;
        double r124851 = r124841 / r124850;
        double r124852 = r124849 + r124851;
        double r124853 = r124836 + r124852;
        double r124854 = a;
        double r124855 = sin(r124854);
        double r124856 = cos(r124854);
        double r124857 = r124855 / r124856;
        double r124858 = r124853 - r124857;
        return r124858;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.1

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

  3. Applied tan-sum0.2

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

  5. Applied sub-neg0.2

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

  6. Applied associate-+r+0.2

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

  8. Applied flip-+0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Taylor expanded around inf 0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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))))