Average Error: 13.2 → 0.2
Time: 42.9s
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\left(\cos a \cdot \left(\cos z \cdot \sin y + \sin z \cdot \cos y\right)\right) \cdot \left(1 + \tan z \cdot \tan y\right) - \left(\sin a \cdot \left(1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos z\right)}{\left(\left(1 + \tan z \cdot \tan y\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) \cdot \cos a\right)} + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\frac{\left(\cos a \cdot \left(\cos z \cdot \sin y + \sin z \cdot \cos y\right)\right) \cdot \left(1 + \tan z \cdot \tan y\right) - \left(\sin a \cdot \left(1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos z\right)}{\left(\left(1 + \tan z \cdot \tan y\right) \cdot \left(\cos y \cdot \cos z\right)\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) \cdot \cos a\right)} + x
double f(double x, double y, double z, double a) {
        double r4715173 = x;
        double r4715174 = y;
        double r4715175 = z;
        double r4715176 = r4715174 + r4715175;
        double r4715177 = tan(r4715176);
        double r4715178 = a;
        double r4715179 = tan(r4715178);
        double r4715180 = r4715177 - r4715179;
        double r4715181 = r4715173 + r4715180;
        return r4715181;
}
↓
double f(double x, double y, double z, double a) {
        double r4715182 = a;
        double r4715183 = cos(r4715182);
        double r4715184 = z;
        double r4715185 = cos(r4715184);
        double r4715186 = y;
        double r4715187 = sin(r4715186);
        double r4715188 = r4715185 * r4715187;
        double r4715189 = sin(r4715184);
        double r4715190 = cos(r4715186);
        double r4715191 = r4715189 * r4715190;
        double r4715192 = r4715188 + r4715191;
        double r4715193 = r4715183 * r4715192;
        double r4715194 = 1.0;
        double r4715195 = tan(r4715184);
        double r4715196 = tan(r4715186);
        double r4715197 = r4715195 * r4715196;
        double r4715198 = r4715194 + r4715197;
        double r4715199 = r4715193 * r4715198;
        double r4715200 = sin(r4715182);
        double r4715201 = r4715197 * r4715197;
        double r4715202 = r4715194 - r4715201;
        double r4715203 = r4715200 * r4715202;
        double r4715204 = r4715190 * r4715185;
        double r4715205 = r4715203 * r4715204;
        double r4715206 = r4715199 - r4715205;
        double r4715207 = r4715198 * r4715204;
        double r4715208 = r4715194 - r4715197;
        double r4715209 = r4715208 * r4715183;
        double r4715210 = r4715207 * r4715209;
        double r4715211 = r4715206 / r4715210;
        double r4715212 = x;
        double r4715213 = r4715211 + r4715212;
        return r4715213;
}

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 13.2

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

  3. Applied tan-quot13.2

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  7. Applied flip--0.2

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

  8. Applied associate-*l/0.2

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

  9. Applied tan-quot0.2

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

  10. Applied tan-quot0.2

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

  11. Applied frac-add0.2

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

  12. Applied associate-*l/0.2

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

  13. Applied frac-sub0.3

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

  14. Applied associate-/l/0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019143 +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))))