Average Error: 13.1 → 0.3
Time: 16.4s
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 r65993 = x;
        double r65994 = y;
        double r65995 = z;
        double r65996 = r65994 + r65995;
        double r65997 = tan(r65996);
        double r65998 = a;
        double r65999 = tan(r65998);
        double r66000 = r65997 - r65999;
        double r66001 = r65993 + r66000;
        return r66001;
}
↓
double f(double x, double y, double z, double a) {
        double r66002 = x;
        double r66003 = y;
        double r66004 = sin(r66003);
        double r66005 = 1.0;
        double r66006 = z;
        double r66007 = sin(r66006);
        double r66008 = r66004 * r66007;
        double r66009 = cos(r66006);
        double r66010 = cos(r66003);
        double r66011 = r66009 * r66010;
        double r66012 = r66008 / r66011;
        double r66013 = r66005 - r66012;
        double r66014 = r66013 * r66010;
        double r66015 = r66004 / r66014;
        double r66016 = r66013 * r66009;
        double r66017 = r66007 / r66016;
        double r66018 = r66015 + r66017;
        double r66019 = r66002 + r66018;
        double r66020 = a;
        double r66021 = sin(r66020);
        double r66022 = cos(r66020);
        double r66023 = r66021 / r66022;
        double r66024 = r66019 - r66023;
        return r66024;
}

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 associate-+r-0.2

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

  7. Applied flip--0.4

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

  8. Taylor expanded around inf 0.3

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

  9. 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))))