Average Error: 13.0 → 0.2
Time: 42.5s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \frac{\mathsf{fma}\left(\cos a, \tan y + \tan z, \sin a \cdot \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)}{\cos a \cdot \left(1 - \frac{{\sin y}^{2}}{{\cos z}^{2}} \cdot \frac{{\sin z}^{2}}{{\cos y}^{2}}\right)} \cdot \left(1 + \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\mathsf{fma}\left(\cos a, \tan y + \tan z, \sin a \cdot \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)}{\cos a \cdot \left(1 - \frac{{\sin y}^{2}}{{\cos z}^{2}} \cdot \frac{{\sin z}^{2}}{{\cos y}^{2}}\right)} \cdot \left(1 + \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\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) (fma (tan y) (tan z) -1.0)))
    (*
     (cos a)
     (-
      1.0
      (*
       (/ (pow (sin y) 2.0) (pow (cos z) 2.0))
       (/ (pow (sin z) 2.0) (pow (cos y) 2.0))))))
   (+ 1.0 (/ (* (sin y) (sin z)) (* (cos z) (cos y)))))))
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) * fma(tan(y), tan(z), -1.0))) / (cos(a) * (1.0 - ((pow(sin(y), 2.0) / pow(cos(z), 2.0)) * (pow(sin(z), 2.0) / pow(cos(y), 2.0)))))) * (1.0 + ((sin(y) * sin(z)) / (cos(z) * cos(y)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.0

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

  2. Applied tan-quot_binary6413.0

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

  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 flip--_binary640.2

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

  9. Applied associate-*r/_binary640.2

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

  10. Applied associate-/r/_binary640.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2021275 
(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))))