Average Error: 13.3 → 0.2
Time: 44.2s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[[y, z] = \mathsf{sort}([y, z]) \\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[\begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\frac{\tan y + \tan z}{1 - \frac{t_0 \cdot \left(\tan y \cdot \sin z\right)}{\cos z}} \cdot \left(1 + t_0\right) - \tan a\right) \end{array} \]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\begin{array}{l}
t_0 := \tan y \cdot \tan z\\
x + \left(\frac{\tan y + \tan z}{1 - \frac{t_0 \cdot \left(\tan y \cdot \sin z\right)}{\cos z}} \cdot \left(1 + t_0\right) - \tan a\right)
\end{array}

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))))
   (+
    x
    (-
     (*
      (/ (+ (tan y) (tan z)) (- 1.0 (/ (* t_0 (* (tan y) (sin z))) (cos z))))
      (+ 1.0 t_0))
     (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) {
	double t_0 = tan(y) * tan(z);
	return x + ((((tan(y) + tan(z)) / (1.0 - ((t_0 * (tan(y) * sin(z))) / cos(z)))) * (1.0 + t_0)) - 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 13.3

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

  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

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

  4. Applied associate-/r/_binary640.2

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

  5. Simplified0.2

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

  6. Applied tan-quot_binary640.2

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

  7. Applied associate-*r/_binary640.2

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

  8. Applied associate-*r/_binary640.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2022097 
(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))))