Average Error: 13.4 → 0.2
Time: 12.5s
Precision: binary64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\frac{\left(\tan y + \tan z\right) \cdot \cos a}{1 - \frac{\sin y \cdot \tan z}{\cos y}} - \sin a}{\cos a} + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\frac{\frac{\left(\tan y + \tan z\right) \cdot \cos a}{1 - \frac{\sin y \cdot \tan z}{\cos y}} - \sin a}{\cos a} + x
double code(double x, double y, double z, double a) {
	return ((double) (x + ((double) (((double) tan(((double) (y + z)))) - ((double) tan(a))))));
}
↓
double code(double x, double y, double z, double a) {
	return ((double) ((((double) ((((double) (((double) (((double) tan(y)) + ((double) tan(z)))) * ((double) cos(a)))) / ((double) (1.0 - (((double) (((double) sin(y)) * ((double) tan(z)))) / ((double) cos(y)))))) - ((double) sin(a)))) / ((double) cos(a))) + x));
}

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 13.4

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-quot13.4

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

  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 associate-/r*0.2

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

  8. Simplified0.2

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

  10. Applied tan-quot0.2

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

  11. Applied associate-*l/0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z a)
  :name "tan-example"
  :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))))