Average Error: 13.2 → 0.2
Time: 32.4s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)}\\ x + \left(\mathsf{fma}\left(\tan y, t_0, \tan z \cdot t_0\right) - \tan a\right) \end{array} \]
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma (tan z) (- (tan y)) 1.0))))
   (+ x (- (fma (tan y) t_0 (* (tan z) 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 = 1.0 / fma(tan(z), -tan(y), 1.0);
	return x + (fma(tan(y), t_0, (tan(z) * t_0)) - tan(a));
}

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end
↓
function code(x, y, z, a)
	t_0 = Float64(1.0 / fma(tan(z), Float64(-tan(y)), 1.0))
	return Float64(x + Float64(fma(tan(y), t_0, Float64(tan(z) * t_0)) - tan(a)))
end

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(N[Tan[z], $MachinePrecision] * (-N[Tan[y], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[Tan[y], $MachinePrecision] * t$95$0 + N[(N[Tan[z], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\begin{array}{l}
t_0 := \frac{1}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)}\\
x + \left(\mathsf{fma}\left(\tan y, t_0, \tan z \cdot t_0\right) - \tan a\right)
\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.2

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

  2. Applied egg-rr0.2

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022166 
(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))))