Average Error: 13.7 → 0.2
Time: 43.0s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \left(\frac{\frac{{\tan y}^{2} - \tan z \cdot \tan z}{\tan y - \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\frac{\frac{{\tan y}^{2} - \tan z \cdot \tan z}{\tan y - \tan z}}{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
  (-
   (/
    (/ (- (pow (tan y) 2.0) (* (tan z) (tan z))) (- (tan y) (tan z)))
    (- 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 + ((((pow(tan(y), 2.0) - (tan(z) * tan(z))) / (tan(y) - tan(z))) / (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 13.7

    \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

  2. Applied tan-sum_binary640.2

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

  3. Applied flip-+_binary640.2

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

  4. Applied pow2_binary640.2

    \[\leadsto x + \left(\frac{\frac{\color{blue}{{\tan y}^{2}} - \tan z \cdot \tan z}{\tan y - \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]

  5. Final simplification0.2

    \[\leadsto x + \left(\frac{\frac{{\tan y}^{2} - \tan z \cdot \tan z}{\tan y - \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]

Reproduce

herbie shell --seed 2021344 
(FPCore (x y z a)
  :name "tan-example"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))