Average Error: 13.2 → 0.2
Time: 29.2s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) - \frac{\sin a}{\frac{\cos a}{1 - \tan y \cdot \tan z}}}{1 - \tan y \cdot \tan z}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\tan y + \tan z\right) - \frac{\sin a}{\frac{\cos a}{1 - \tan y \cdot \tan z}}}{1 - \tan y \cdot \tan z}
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (/
   (- (+ (tan y) (tan z)) (/ (sin a) (/ (cos a) (- 1.0 (* (tan y) (tan z))))))
   (- 1.0 (* (tan y) (tan z))))))
double code(double x, double y, double z, double a) {
	return x + (tan(y + z) - tan(a));
}
↓
double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) - (sin(a) / (cos(a) / (1.0 - (tan(y) * tan(z)))))) / (1.0 - (tan(y) * tan(z))));
}

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 13.2

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

  3. Applied tan-quot_binary6413.2

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

  4. Applied tan-sum_binary640.2

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

  5. Applied frac-sub_binary640.2

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  9. Applied associate-/r*_binary640.2

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

  10. Simplified0.2

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

  12. Applied associate-/l*_binary640.2

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

  13. Final simplification0.2

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

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))