Average Error: 13.5 → 0.2
Time: 38.8s
Precision: binary64
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\frac{\sin y \cdot \sin z}{\cos y \cdot \cos z}, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right) \]
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (+
   (fma
    (/ (+ (tan y) (tan z)) (- 1.0 (pow (* (tan y) (tan z)) 3.0)))
    (fma
     (/ (* (sin y) (sin z)) (* (cos y) (cos z)))
     (fma (tan y) (tan z) 1.0)
     1.0)
    (- (tan a)))
   (* (tan a) 0.0))))
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(((tan(y) + tan(z)) / (1.0 - pow((tan(y) * tan(z)), 3.0))), fma(((sin(y) * sin(z)) / (cos(y) * cos(z))), fma(tan(y), tan(z), 1.0), 1.0), -tan(a)) + (tan(a) * 0.0));
}

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 + Float64(fma(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - (Float64(tan(y) * tan(z)) ^ 3.0))), fma(Float64(Float64(sin(y) * sin(z)) / Float64(cos(y) * cos(z))), fma(tan(y), tan(z), 1.0), 1.0), Float64(-tan(a))) + Float64(tan(a) * 0.0)))
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[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision] + N[(N[Tan[a], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\frac{\sin y \cdot \sin z}{\cos y \cdot \cos z}, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.5

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

  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 add-cube-cbrt_binary640.3

    \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right) \cdot \sqrt[3]{\tan a}}\right) \]

  4. Applied flip3--_binary640.3

    \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\frac{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}{1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)}}} - \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right) \cdot \sqrt[3]{\tan a}\right) \]

  5. Applied associate-/r/_binary640.3

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)} - \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right) \cdot \sqrt[3]{\tan a}\right) \]

  6. Applied prod-diff_binary640.3

    \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right), -\sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)\right)} \]

  7. Simplified0.2

    \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\tan y \cdot \tan z, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right)} + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)\right) \]

  8. Simplified0.2

    \[\leadsto x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\tan y \cdot \tan z, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \color{blue}{\tan a \cdot 0}\right) \]

  9. Taylor expanded in y around inf 0.2

    \[\leadsto x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\color{blue}{\frac{\sin y \cdot \sin z}{\cos y \cdot \cos z}}, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right) \]

  10. Final simplification0.2

    \[\leadsto x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\frac{\sin y \cdot \sin z}{\cos y \cdot \cos z}, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right) \]

Reproduce

herbie shell --seed 2022129 
(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))))