Average Error: 13.0 → 0.2
Time: 37.7s
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{\cos z \cdot \sin y + \sin z \cdot \cos y}{\left(\cos y \cdot \cos z\right) \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)} - \tan a\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\frac{\cos z \cdot \sin y + \sin z \cdot \cos y}{\left(\cos y \cdot \cos z\right) \cdot \left(1 - \frac{\sin z \cdot \sin y}{\cos y \cdot \cos z}\right)} - \tan a\right)
double f(double x, double y, double z, double a) {
        double r3461914 = x;
        double r3461915 = y;
        double r3461916 = z;
        double r3461917 = r3461915 + r3461916;
        double r3461918 = tan(r3461917);
        double r3461919 = a;
        double r3461920 = tan(r3461919);
        double r3461921 = r3461918 - r3461920;
        double r3461922 = r3461914 + r3461921;
        return r3461922;
}
↓
double f(double x, double y, double z, double a) {
        double r3461923 = x;
        double r3461924 = z;
        double r3461925 = cos(r3461924);
        double r3461926 = y;
        double r3461927 = sin(r3461926);
        double r3461928 = r3461925 * r3461927;
        double r3461929 = sin(r3461924);
        double r3461930 = cos(r3461926);
        double r3461931 = r3461929 * r3461930;
        double r3461932 = r3461928 + r3461931;
        double r3461933 = r3461930 * r3461925;
        double r3461934 = 1.0;
        double r3461935 = r3461929 * r3461927;
        double r3461936 = r3461935 / r3461933;
        double r3461937 = r3461934 - r3461936;
        double r3461938 = r3461933 * r3461937;
        double r3461939 = r3461932 / r3461938;
        double r3461940 = a;
        double r3461941 = tan(r3461940);
        double r3461942 = r3461939 - r3461941;
        double r3461943 = r3461923 + r3461942;
        return r3461943;
}

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 13.0

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 tan-quot0.2

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

  6. Applied tan-quot0.2

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

  7. Applied frac-add0.2

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

  8. Applied associate-/l/0.2

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

  9. Taylor expanded around inf 0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 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))))