Average Error: 12.9 → 0.2
Time: 50.3s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{\frac{\cos z \cdot \sin y + \sin z \cdot \cos y}{\cos z \cdot \cos y}}{1 - \tan y \cdot \tan z} - \tan a\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\frac{\frac{\cos z \cdot \sin y + \sin z \cdot \cos y}{\cos z \cdot \cos y}}{1 - \tan y \cdot \tan z} - \tan a\right)
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/
    (/ (+ (* (cos z) (sin y)) (* (sin z) (cos y))) (* (cos z) (cos y)))
    (- 1.0 (* (tan y) (tan z))))
   (tan a))))
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 + (((((cos(z) * sin(y)) + (sin(z) * cos(y))) / (cos(z) * cos(y))) / (1.0 - (tan(y) * tan(z)))) - tan(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 12.9

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

  3. Applied tan-sum_binary640.2

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

  5. Applied tan-quot_binary640.2

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

  6. Applied tan-quot_binary640.2

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

  7. Applied frac-add_binary640.2

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2021090 
(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))))