Average Error: 12.9 → 0.2
Time: 8.1s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a + \sin a \cdot \left(\tan y \cdot \tan z + -1\right)}{\cos a \cdot \left(1 - \frac{\tan z \cdot \sin y}{\cos y}\right)}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\tan y + \tan z\right) \cdot \cos a + \sin a \cdot \left(\tan y \cdot \tan z + -1\right)}{\cos a \cdot \left(1 - \frac{\tan z \cdot \sin y}{\cos y}\right)}
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) (x + (((double) (((double) (((double) (((double) tan(y)) + ((double) tan(z)))) * ((double) cos(a)))) + ((double) (((double) sin(a)) * ((double) (((double) (((double) tan(y)) * ((double) tan(z)))) + -1.0)))))) / ((double) (((double) cos(a)) * ((double) (1.0 - (((double) (((double) tan(z)) * ((double) sin(y)))) / ((double) cos(y))))))))));
}

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-quot12.9

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

  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. Simplified0.2

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

  8. Applied tan-quot0.2

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

  9. Applied associate-*l/0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020196 
(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))))