Average Error: 13.1 → 0.3
Time: 15.5s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(x + \left(\frac{\sin y}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos y} + \frac{\sin z}{\left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \cos z}\right)\right) - \frac{\sin a}{\cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left(x + \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)\right) - \frac{\sin a}{\cos a}
double f(double x, double y, double z, double a) {
        double r59326 = x;
        double r59327 = y;
        double r59328 = z;
        double r59329 = r59327 + r59328;
        double r59330 = tan(r59329);
        double r59331 = a;
        double r59332 = tan(r59331);
        double r59333 = r59330 - r59332;
        double r59334 = r59326 + r59333;
        return r59334;
}
↓
double f(double x, double y, double z, double a) {
        double r59335 = x;
        double r59336 = y;
        double r59337 = sin(r59336);
        double r59338 = 1.0;
        double r59339 = z;
        double r59340 = sin(r59339);
        double r59341 = r59337 * r59340;
        double r59342 = cos(r59339);
        double r59343 = cos(r59336);
        double r59344 = r59342 * r59343;
        double r59345 = r59341 / r59344;
        double r59346 = r59338 - r59345;
        double r59347 = r59346 * r59343;
        double r59348 = r59337 / r59347;
        double r59349 = r59346 * r59342;
        double r59350 = r59340 / r59349;
        double r59351 = r59348 + r59350;
        double r59352 = r59335 + r59351;
        double r59353 = a;
        double r59354 = sin(r59353);
        double r59355 = cos(r59353);
        double r59356 = r59354 / r59355;
        double r59357 = r59352 - r59356;
        return r59357;
}

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-sum0.2

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

  5. Applied sub-neg0.2

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

  6. Applied associate-+r+0.2

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

  8. Applied flip-+0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Taylor expanded around inf 0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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))))