Average Error: 12.8 → 0.2
Time: 58.7s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \frac{\mathsf{fma}\left(\cos a, \tan y + \tan z, \sin a \cdot \left(\tan y \cdot \tan z + -1\right)\right)}{\cos a \cdot \left(1 - \frac{\sin y}{\cos y} \cdot \frac{\sin z}{\cos z}\right)} \]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\mathsf{fma}\left(\cos a, \tan y + \tan z, \sin a \cdot \left(\tan y \cdot \tan z + -1\right)\right)}{\cos a \cdot \left(1 - \frac{\sin y}{\cos y} \cdot \frac{\sin z}{\cos z}\right)}

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (/
   (fma (cos a) (+ (tan y) (tan z)) (* (sin a) (+ (* (tan y) (tan z)) -1.0)))
   (* (cos a) (- 1.0 (* (/ (sin y) (cos y)) (/ (sin z) (cos 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 + (fma(cos(a), (tan(y) + tan(z)), (sin(a) * ((tan(y) * tan(z)) + -1.0))) / (cos(a) * (1.0 - ((sin(y) / cos(y)) * (sin(z) / cos(z))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 12.8

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

  2. Applied tan-quot_binary6412.8

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

  3. Applied tan-sum_binary640.2

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

  4. Applied frac-sub_binary640.2

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

  5. Simplified0.2

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

  6. Simplified0.2

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

  7. Taylor expanded in y around inf 0.2

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

  8. Applied times-frac_binary640.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2021280 
(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))))