Average Error: 12.9 → 0.2
Time: 13.3s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\left(\frac{\sin y}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\left(1 - \frac{\sin y}{\cos z} \cdot \frac{\sin z}{\cos y}\right) \cdot \cos z}\right) - \frac{\sin a}{\cos a}\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\left(\frac{\sin y}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\left(1 - \frac{\sin y}{\cos z} \cdot \frac{\sin z}{\cos y}\right) \cdot \cos z}\right) - \frac{\sin a}{\cos a}\right)
double f(double x, double y, double z, double a) {
        double r172339 = x;
        double r172340 = y;
        double r172341 = z;
        double r172342 = r172340 + r172341;
        double r172343 = tan(r172342);
        double r172344 = a;
        double r172345 = tan(r172344);
        double r172346 = r172343 - r172345;
        double r172347 = r172339 + r172346;
        return r172347;
}
↓
double f(double x, double y, double z, double a) {
        double r172348 = x;
        double r172349 = y;
        double r172350 = sin(r172349);
        double r172351 = 1.0;
        double r172352 = z;
        double r172353 = sin(r172352);
        double r172354 = r172350 * r172353;
        double r172355 = cos(r172352);
        double r172356 = cos(r172349);
        double r172357 = r172355 * r172356;
        double r172358 = r172354 / r172357;
        double r172359 = r172351 - r172358;
        double r172360 = r172359 * r172356;
        double r172361 = r172350 / r172360;
        double r172362 = r172350 / r172355;
        double r172363 = r172353 / r172356;
        double r172364 = r172362 * r172363;
        double r172365 = r172351 - r172364;
        double r172366 = r172365 * r172355;
        double r172367 = r172353 / r172366;
        double r172368 = r172361 + r172367;
        double r172369 = a;
        double r172370 = sin(r172369);
        double r172371 = cos(r172369);
        double r172372 = r172370 / r172371;
        double r172373 = r172368 - r172372;
        double r172374 = r172348 + r172373;
        return r172374;
}

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 12.9

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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. Taylor expanded around inf 0.2

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

  6. Applied times-frac0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020003 +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))))