Average Error: 13.1 → 0.2
Time: 29.9s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\frac{\cos a \cdot \sin z}{\cos z} + \left(\frac{\sin a \cdot \left(\sin z \cdot \sin y\right)}{\cos z \cdot \cos y} + \frac{\cos a \cdot \sin y}{\cos y}\right)\right) - \sin a}{\cos a \cdot \left(1 - \tan y \cdot \tan z\right)}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\frac{\cos a \cdot \sin z}{\cos z} + \left(\frac{\sin a \cdot \left(\sin z \cdot \sin y\right)}{\cos z \cdot \cos y} + \frac{\cos a \cdot \sin y}{\cos y}\right)\right) - \sin a}{\cos a \cdot \left(1 - \tan y \cdot \tan z\right)}
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (/
   (-
    (+
     (/ (* (cos a) (sin z)) (cos z))
     (+
      (/ (* (sin a) (* (sin z) (sin y))) (* (cos z) (cos y)))
      (/ (* (cos a) (sin y)) (cos y))))
    (sin a))
   (* (cos a) (- 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 + (((((cos(a) * sin(z)) / cos(z)) + (((sin(a) * (sin(z) * sin(y))) / (cos(z) * cos(y))) + ((cos(a) * sin(y)) / cos(y)))) - sin(a)) / (cos(a) * (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 13.1

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

  3. Applied tan-quot_binary6413.1

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

  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. Taylor expanded around inf 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020292 
(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))))