Average Error: 13.3 → 0.2
Time: 35.2s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \mathsf{fma}\left(1, \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}, -\tan a\right) \]
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma 1.0 (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- (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) {
	return x + fma(1.0, ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))), -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)
	return Float64(x + fma(1.0, Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))), Float64(-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_] := N[(x + N[(1.0 * N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.3

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

  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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022151 
(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))))