Average Error: 13.4 → 0.2
Time: 41.1s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[\begin{array}{l} t_0 := \tan y \cdot \tan z\\ t_1 := -\tan a\\ \left(x + \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - t_0 \cdot t_0}, 1 + t_0, t_1\right)\right) + \mathsf{fma}\left(t_1, 1, \tan a\right) \end{array} \]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\begin{array}{l}
t_0 := \tan y \cdot \tan z\\
t_1 := -\tan a\\
\left(x + \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - t_0 \cdot t_0}, 1 + t_0, t_1\right)\right) + \mathsf{fma}\left(t_1, 1, \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))) (t_1 (- (tan a))))
   (+
    (+ x (fma (/ (+ (tan y) (tan z)) (- 1.0 (* t_0 t_0))) (+ 1.0 t_0) t_1))
    (fma t_1 1.0 (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);
	double t_1 = -tan(a);
	return (x + fma(((tan(y) + tan(z)) / (1.0 - (t_0 * t_0))), (1.0 + t_0), t_1)) + fma(t_1, 1.0, tan(a));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.4

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

  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 *-un-lft-identity_binary640.2

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

  4. Applied flip--_binary640.2

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

  5. Applied associate-/r/_binary640.2

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

  6. Applied prod-diff_binary640.2

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

  7. Applied associate-+r+_binary640.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2022125 
(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))))