2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.4%

Time: 12.8s

Alternatives: 9

Speedup: 17.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x + {\tan x}^{3}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma (+ (tan x) (pow (tan x) 3.0)) eps (pow (tan x) 2.0)) eps eps))

C

double code(double x, double eps) {
	return fma(fma((tan(x) + pow(tan(x), 3.0)), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(Float64(tan(x) + (tan(x) ^ 3.0)), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]

   2. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(x + \varepsilon\right)} - \tan x \]

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + 1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + \color{blue}{\varepsilon} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x + {\tan x}^{3}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\\ \mathsf{fma}\left(t\_0, {\tan x}^{2}, t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma eps (tan x) 1.0))) (* (fma t_0 (pow (tan x) 2.0) t_0) eps)))

C

double code(double x, double eps) {
	double t_0 = fma(eps, tan(x), 1.0);
	return fma(t_0, pow(tan(x), 2.0), t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = fma(eps, tan(x), 1.0)
	return Float64(fma(t_0, (tan(x) ^ 2.0), t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(t$95$0 * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \tan x, 1\right), {\tan x}^{2}, \mathsf{fma}\left(\varepsilon, \tan x, 1\right) \cdot 1\right) \cdot \varepsilon \]

8. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \tan x, 1\right), {\tan x}^{2}, \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 3: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \left(\mathsf{fma}\left(\varepsilon, \tan x, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (tan x) (tan x) 1.0) (* (fma eps (tan x) 1.0) eps)))

C

double code(double x, double eps) {
	return fma(tan(x), tan(x), 1.0) * (fma(eps, tan(x), 1.0) * eps);
}

Julia

function code(x, eps)
	return Float64(fma(tan(x), tan(x), 1.0) * Float64(fma(eps, tan(x), 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(eps * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon} \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \tan x, 1\right) \cdot \varepsilon\right)} \]
8. Add Preprocessing

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 1.3333333333333333, x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma (fma (pow x 3.0) 1.3333333333333333 x) eps (pow (tan x) 2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(pow(x, 3.0), 1.3333333333333333, x), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma((x ^ 3.0), 1.3333333333333333, x), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * 1.3333333333333333 + x), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]

   2. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(x + \varepsilon\right)} - \tan x \]

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + 1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + \color{blue}{\varepsilon} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x + {\tan x}^{3}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

12. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 1.3333333333333333, x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 5: 98.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, 1.3333333333333333 \cdot \varepsilon\right), x, 1\right), x, \varepsilon\right) \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (fma
    (fma (fma 0.6666666666666666 x (* 1.3333333333333333 eps)) x 1.0)
    x
    eps)
   x)
  eps
  eps))

C

double code(double x, double eps) {
	return fma((fma(fma(fma(0.6666666666666666, x, (1.3333333333333333 * eps)), x, 1.0), x, eps) * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(fma(fma(fma(0.6666666666666666, x, Float64(1.3333333333333333 * eps)), x, 1.0), x, eps) * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.6666666666666666 * x + N[(1.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + eps), $MachinePrecision] * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]

   2. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(x + \varepsilon\right)} - \tan x \]

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + 1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + \color{blue}{\varepsilon} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} - -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon\right)\right)\right), \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, 1.3333333333333333 \cdot \varepsilon\right), x, 1\right), x, \varepsilon\right) \cdot x, \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 6: 98.3% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), x, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* eps (+ eps x)) x eps))

C

double code(double x, double eps) {
	return fma((eps * (eps + x)), x, eps);
}

Julia

function code(x, eps)
	return fma(Float64(eps * Float64(eps + x)), x, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * x + eps), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{x \cdot \left(\varepsilon \cdot x + {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), \color{blue}{x}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 7: 97.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* eps eps) x eps))

C

double code(double x, double eps) {
	return fma((eps * eps), x, eps);
}

Julia

function code(x, eps)
	return fma(Float64(eps * eps), x, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * x + eps), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{\varepsilon}^{2} \cdot x} \]
7. Step-by-step derivation

8. Applied rewrites 97.2%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{x}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 8: 5.7% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) x))

C

double code(double x, double eps) {
	return (eps * eps) * x;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * x
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * x;
}

Python

def code(x, eps):
	return (eps * eps) * x

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * x)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * x;
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{\varepsilon}^{2} \cdot x} \]
7. Step-by-step derivation

8. Applied rewrites 97.2%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{x}, \varepsilon\right) \]

9. Taylor expanded in x around inf

   \[\leadsto {\varepsilon}^{2} \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 5.6%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot x \]
12. Add Preprocessing

Alternative 9: 5.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 60.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right) - \tan x} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right) + \left(\mathsf{neg}\left(\tan x\right)\right)} \]

   3. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} + \left(\mathsf{neg}\left(\tan x\right)\right) \]

   4. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} + \left(\mathsf{neg}\left(\tan x\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin \left(x + \varepsilon\right)\right)}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}} + \left(\mathsf{neg}\left(\tan x\right)\right) \]

   6. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \left(x + \varepsilon\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}} + \left(\mathsf{neg}\left(\tan x\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\sin \left(x + \varepsilon\right)\right), \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\sin \left(x + \varepsilon\right)}, \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right) \]

   9. lower-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(-\color{blue}{\sin \left(x + \varepsilon\right)}, \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \color{blue}{\left(x + \varepsilon\right)}, \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(-\sin \color{blue}{\left(\varepsilon + x\right)}, \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \color{blue}{\left(\varepsilon + x\right)}, \frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}, \mathsf{neg}\left(\tan x\right)\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \color{blue}{\frac{1}{\mathsf{neg}\left(\cos \left(x + \varepsilon\right)\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   14. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{\color{blue}{-\cos \left(x + \varepsilon\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   15. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{-\color{blue}{\cos \left(x + \varepsilon\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{-\cos \color{blue}{\left(x + \varepsilon\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{-\cos \color{blue}{\left(\varepsilon + x\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   18. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{-\cos \color{blue}{\left(\varepsilon + x\right)}}, \mathsf{neg}\left(\tan x\right)\right) \]

   19. lower-neg.f64 60.5

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

4. Applied rewrites 60.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\sin \left(\varepsilon + x\right), \frac{1}{-\cos \left(\varepsilon + x\right)}, -\tan x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(-\sin \left(\varepsilon + x\right)\right) \cdot \frac{1}{-\cos \left(\varepsilon + x\right)} + \left(-\tan x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{-\cos \left(\varepsilon + x\right)} \cdot \left(-\sin \left(\varepsilon + x\right)\right)} + \left(-\tan x\right) \]

   3. lower-fma.f64 60.5

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{-\cos \left(\varepsilon + x\right)}, -\sin \left(\varepsilon + x\right), -\tan x\right) \]

   6. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(\cos \left(\varepsilon + x\right)\right)}}, -\sin \left(\varepsilon + x\right), -\tan x\right) \]

   7. frac-2neg N/A

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

   8. lower-/.f64 60.5

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

6. Applied rewrites 60.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\cos \left(\varepsilon + x\right)}, -\sin \left(\varepsilon + x\right), -\tan x\right)} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \left(-\sin \left(\varepsilon + x\right)\right) + \left(-\tan x\right)} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\varepsilon + x\right)\right)\right)} + \left(-\tan x\right) \]

   3. distribute-rgt-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \sin \left(\varepsilon + x\right)\right)\right)} + \left(-\tan x\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \sin \left(\varepsilon + x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\tan x\right)\right)} \]

   5. distribute-neg-out N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \sin \left(\varepsilon + x\right) + \tan x\right)\right)} \]

   6. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\left(\frac{-1}{\cos \left(\varepsilon + x\right)} \cdot \sin \left(\varepsilon + x\right) + \tan x\right)} \]

   7. *-commutative N/A

      \[\leadsto -\left(\color{blue}{\sin \left(\varepsilon + x\right) \cdot \frac{-1}{\cos \left(\varepsilon + x\right)}} + \tan x\right) \]

   8. lower-fma.f64 60.5

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

8. Applied rewrites 60.5%

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

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right)} \]
10. Step-by-step derivation

    1. distribute-lft1-in N/A

       \[\leadsto -1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{\sin x}{\cos x}\right)} \]

    2. metadata-eval N/A

       \[\leadsto -1 \cdot \left(\color{blue}{0} \cdot \frac{\sin x}{\cos x}\right) \]

    3. mul0-lft N/A

       \[\leadsto -1 \cdot \color{blue}{0} \]

    4. metadata-eval 5.2

       \[\leadsto \color{blue}{0} \]

11. Applied rewrites 5.2%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \left(\varepsilon \cdot \tan x\right) \cdot \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* (* eps (tan x)) (tan x))))

C

double code(double x, double eps) {
	return eps + ((eps * tan(x)) * tan(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((eps * tan(x)) * tan(x))
end function

Java

public static double code(double x, double eps) {
	return eps + ((eps * Math.tan(x)) * Math.tan(x));
}

Python

def code(x, eps):
	return eps + ((eps * math.tan(x)) * math.tan(x))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(eps * tan(x)) * tan(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((eps * tan(x)) * tan(x));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(eps * N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024296 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))