Average Error: 13.3 → 0.2
Time: 24.8s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{\sin y}{\cos y \cdot \left(1 - \sin z \cdot \frac{\sin y}{\cos y \cdot \cos z}\right)} + \left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\sin z}{\cos y} \cdot \frac{\sin y}{\cos z}\right)} - \frac{\sin a}{\cos a}\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\frac{\sin y}{\cos y \cdot \left(1 - \sin z \cdot \frac{\sin y}{\cos y \cdot \cos z}\right)} + \left(\frac{\sin z}{\cos z \cdot \left(1 - \frac{\sin z}{\cos y} \cdot \frac{\sin y}{\cos z}\right)} - \frac{\sin a}{\cos a}\right)\right)
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (+
   (/ (sin y) (* (cos y) (- 1.0 (* (sin z) (/ (sin y) (* (cos y) (cos z)))))))
   (-
    (/ (sin z) (* (cos z) (- 1.0 (* (/ (sin z) (cos y)) (/ (sin y) (cos z))))))
    (/ (sin a) (cos a))))))
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) sin(y)) / ((double) (((double) cos(y)) * ((double) (1.0 - ((double) (((double) sin(z)) * (((double) sin(y)) / ((double) (((double) cos(y)) * ((double) cos(z)))))))))))) + ((double) ((((double) sin(z)) / ((double) (((double) cos(z)) * ((double) (1.0 - ((double) ((((double) sin(z)) / ((double) cos(y))) * (((double) sin(y)) / ((double) cos(z)))))))))) - (((double) sin(a)) / ((double) cos(a)))))))));
}

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 Error: 13.3 bits

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

  3. Applied tan-sumError: 0.2 bits

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

  4. Taylor expanded around inf Error: 0.2 bits

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

  5. SimplifiedError: 0.2 bits

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

  7. Applied *-un-lft-identityError: 0.2 bits

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

  8. Applied times-fracError: 0.2 bits

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

  9. Applied associate-*r*Error: 0.2 bits

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

  10. SimplifiedError: 0.2 bits

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

  11. Final simplificationError: 0.2 bits

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

Reproduce

herbie shell --seed 2020204 
(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))))