Average Error: 13.1 → 0.2
Time: 12.5s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{1 - \left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right)}}{\cos a} \cdot \left(1 + \tan y \cdot \tan z\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{1 - \left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right)}}{\cos a} \cdot \left(1 + \tan y \cdot \tan z\right)
double f(double x, double y, double z, double a) {
        double r164481 = x;
        double r164482 = y;
        double r164483 = z;
        double r164484 = r164482 + r164483;
        double r164485 = tan(r164484);
        double r164486 = a;
        double r164487 = tan(r164486);
        double r164488 = r164485 - r164487;
        double r164489 = r164481 + r164488;
        return r164489;
}
↓
double f(double x, double y, double z, double a) {
        double r164490 = x;
        double r164491 = y;
        double r164492 = tan(r164491);
        double r164493 = z;
        double r164494 = tan(r164493);
        double r164495 = r164492 + r164494;
        double r164496 = a;
        double r164497 = cos(r164496);
        double r164498 = r164495 * r164497;
        double r164499 = 1.0;
        double r164500 = r164492 * r164494;
        double r164501 = r164499 - r164500;
        double r164502 = sin(r164496);
        double r164503 = r164501 * r164502;
        double r164504 = r164498 - r164503;
        double r164505 = r164500 * r164500;
        double r164506 = r164499 - r164505;
        double r164507 = r164504 / r164506;
        double r164508 = r164507 / r164497;
        double r164509 = r164499 + r164500;
        double r164510 = r164508 * r164509;
        double r164511 = r164490 + r164510;
        return r164511;
}

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-quot13.1

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

  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

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

  8. Applied associate-*l/0.2

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

  9. Applied associate-/r/0.2

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

  10. Simplified0.2

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

  12. Applied associate-/r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 +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))))