Average Error: 13.0 → 0.2
Time: 9.0s
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 r119361 = x;
        double r119362 = y;
        double r119363 = z;
        double r119364 = r119362 + r119363;
        double r119365 = tan(r119364);
        double r119366 = a;
        double r119367 = tan(r119366);
        double r119368 = r119365 - r119367;
        double r119369 = r119361 + r119368;
        return r119369;
}
↓
double f(double x, double y, double z, double a) {
        double r119370 = x;
        double r119371 = y;
        double r119372 = tan(r119371);
        double r119373 = z;
        double r119374 = tan(r119373);
        double r119375 = r119372 + r119374;
        double r119376 = a;
        double r119377 = cos(r119376);
        double r119378 = r119375 * r119377;
        double r119379 = 1.0;
        double r119380 = r119372 * r119374;
        double r119381 = r119379 - r119380;
        double r119382 = sin(r119376);
        double r119383 = r119381 * r119382;
        double r119384 = r119378 - r119383;
        double r119385 = sin(r119371);
        double r119386 = r119385 * r119374;
        double r119387 = cos(r119371);
        double r119388 = r119386 / r119387;
        double r119389 = r119379 - r119388;
        double r119390 = r119389 * r119377;
        double r119391 = r119384 / r119390;
        double r119392 = r119370 + r119391;
        return r119392;
}

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))))