Average Error: 13.0 → 0.2
Time: 11.0s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \frac{\sin y \cdot \tan z}{\cos y}\right) \cdot \cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \frac{\sin y \cdot \tan z}{\cos y}\right) \cdot \cos a}
double f(double x, double y, double z, double a) {
        double r164977 = x;
        double r164978 = y;
        double r164979 = z;
        double r164980 = r164978 + r164979;
        double r164981 = tan(r164980);
        double r164982 = a;
        double r164983 = tan(r164982);
        double r164984 = r164981 - r164983;
        double r164985 = r164977 + r164984;
        return r164985;
}
↓
double f(double x, double y, double z, double a) {
        double r164986 = x;
        double r164987 = y;
        double r164988 = tan(r164987);
        double r164989 = z;
        double r164990 = tan(r164989);
        double r164991 = r164988 + r164990;
        double r164992 = a;
        double r164993 = cos(r164992);
        double r164994 = r164991 * r164993;
        double r164995 = 1.0;
        double r164996 = r164988 * r164990;
        double r164997 = r164995 - r164996;
        double r164998 = sin(r164992);
        double r164999 = r164997 * r164998;
        double r165000 = r164994 - r164999;
        double r165001 = sin(r164987);
        double r165002 = r165001 * r164990;
        double r165003 = cos(r164987);
        double r165004 = r165002 / r165003;
        double r165005 = r164995 - r165004;
        double r165006 = r165005 * r164993;
        double r165007 = r165000 / r165006;
        double r165008 = r164986 + r165007;
        return r165008;
}

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-quot13.0

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

  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 tan-quot0.2

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

  8. Applied associate-*l/0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 
(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))))