2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.8%

Time: 17.6s

Alternatives: 14

Speedup: 205.0×

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: 62.2% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon \cdot \tan x\\ \frac{\tan x \cdot \left(\frac{\tan \varepsilon}{\tan x} + t\_0\right)}{1 - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan eps) (tan x))))
   (/ (* (tan x) (+ (/ (tan eps) (tan x)) t_0)) (- 1.0 t_0))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) * tan(x);
	return (tan(x) * ((tan(eps) / tan(x)) + t_0)) / (1.0 - t_0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(eps) * tan(x)
    code = (tan(x) * ((tan(eps) / tan(x)) + t_0)) / (1.0d0 - t_0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) * Math.tan(x);
	return (Math.tan(x) * ((Math.tan(eps) / Math.tan(x)) + t_0)) / (1.0 - t_0);
}

Python

def code(x, eps):
	t_0 = math.tan(eps) * math.tan(x)
	return (math.tan(x) * ((math.tan(eps) / math.tan(x)) + t_0)) / (1.0 - t_0)

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) * tan(x))
	return Float64(Float64(tan(x) * Float64(Float64(tan(eps) / tan(x)) + t_0)) / Float64(1.0 - t_0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(eps) * tan(x);
	tmp = (tan(x) * ((tan(eps) / tan(x)) + t_0)) / (1.0 - t_0);
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

   1. tan-sum N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1\right), \color{blue}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}\right)}\right) \]

4. Applied egg-rr 60.4%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \left(\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x} + \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)\right), \color{blue}{\left(\cos x \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)\right)}\right) \]

7. Simplified 99.7%

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. frac-times N/A

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

   5. tan-quot N/A

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

   6. tan-quot N/A

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

   7. *-commutative N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(\frac{\cos x \cdot \sin \varepsilon}{\sin x \cdot \cos \varepsilon} + \sin \varepsilon \cdot \frac{\sin x}{\cos x \cdot \cos \varepsilon}\right)}{\cos x}\right), \color{blue}{\left(1 - \tan x \cdot \tan \varepsilon\right)}\right) \]

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

    \[\leadsto \frac{\tan x \cdot \left(\frac{\tan \varepsilon}{\tan x} + \tan \varepsilon \cdot \tan x\right)}{1 - \tan \varepsilon \cdot \tan x} \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x \cdot \left(\tan \varepsilon \cdot \left(\tan x + \frac{1}{\tan x}\right)\right)}{1 - \tan \varepsilon \cdot \tan x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (tan x) (* (tan eps) (+ (tan x) (/ 1.0 (tan x)))))
  (- 1.0 (* (tan eps) (tan x)))))

C

double code(double x, double eps) {
	return (tan(x) * (tan(eps) * (tan(x) + (1.0 / tan(x))))) / (1.0 - (tan(eps) * tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (tan(x) * (tan(eps) * (tan(x) + (1.0d0 / tan(x))))) / (1.0d0 - (tan(eps) * tan(x)))
end function

Java

public static double code(double x, double eps) {
	return (Math.tan(x) * (Math.tan(eps) * (Math.tan(x) + (1.0 / Math.tan(x))))) / (1.0 - (Math.tan(eps) * Math.tan(x)));
}

Python

def code(x, eps):
	return (math.tan(x) * (math.tan(eps) * (math.tan(x) + (1.0 / math.tan(x))))) / (1.0 - (math.tan(eps) * math.tan(x)))

Julia

function code(x, eps)
	return Float64(Float64(tan(x) * Float64(tan(eps) * Float64(tan(x) + Float64(1.0 / tan(x))))) / Float64(1.0 - Float64(tan(eps) * tan(x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (tan(x) * (tan(eps) * (tan(x) + (1.0 / tan(x))))) / (1.0 - (tan(eps) * tan(x)));
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

   1. tan-sum N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1\right), \color{blue}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}\right)}\right) \]

4. Applied egg-rr 60.4%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \left(\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x} + \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)\right), \color{blue}{\left(\cos x \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)\right)}\right) \]

7. Simplified 99.7%

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. frac-times N/A

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

   5. tan-quot N/A

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

   6. tan-quot N/A

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

   7. *-commutative N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(\frac{\cos x \cdot \sin \varepsilon}{\sin x \cdot \cos \varepsilon} + \sin \varepsilon \cdot \frac{\sin x}{\cos x \cdot \cos \varepsilon}\right)}{\cos x}\right), \color{blue}{\left(1 - \tan x \cdot \tan \varepsilon\right)}\right) \]

9. Applied egg-rr 99.8%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \frac{\tan x \cdot \left(\tan \varepsilon \cdot \left(\tan x + \frac{1}{\tan x}\right)\right)}{1 - \tan \varepsilon \cdot \tan x} \]
13. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \tan x \cdot \frac{\frac{\varepsilon}{\sin x \cdot \cos x}}{1 - \tan \varepsilon \cdot \tan x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (tan x) (/ (/ eps (* (sin x) (cos x))) (- 1.0 (* (tan eps) (tan x))))))

C

double code(double x, double eps) {
	return tan(x) * ((eps / (sin(x) * cos(x))) / (1.0 - (tan(eps) * tan(x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan(x) * ((eps / (sin(x) * cos(x))) / (1.0d0 - (tan(eps) * tan(x))))
end function

Java

public static double code(double x, double eps) {
	return Math.tan(x) * ((eps / (Math.sin(x) * Math.cos(x))) / (1.0 - (Math.tan(eps) * Math.tan(x))));
}

Python

def code(x, eps):
	return math.tan(x) * ((eps / (math.sin(x) * math.cos(x))) / (1.0 - (math.tan(eps) * math.tan(x))))

Julia

function code(x, eps)
	return Float64(tan(x) * Float64(Float64(eps / Float64(sin(x) * cos(x))) / Float64(1.0 - Float64(tan(eps) * tan(x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan(x) * ((eps / (sin(x) * cos(x))) / (1.0 - (tan(eps) * tan(x))));
end

Wolfram

code[x_, eps_] := N[(N[Tan[x], $MachinePrecision] * N[(N[(eps / N[(N[Sin[x], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

   1. tan-sum N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1\right), \color{blue}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}\right)}\right) \]

4. Applied egg-rr 60.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{\mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{\sin x}{\cos x}\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{\mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin x, \cos x\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\mathsf{tan.f64}\left(x\right)}, \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \cos x\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(\color{blue}{x}\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\cos x, \sin x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{\mathsf{tan.f64}\left(\varepsilon\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \sin x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\color{blue}{\varepsilon}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. sin-lowering-sin.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{sin.f64}\left(x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

7. Simplified 99.1%

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

   1. un-div-inv N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right)}{1 - \tan x \cdot \tan \varepsilon}\right), \color{blue}{\tan x}\right) \]

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \tan x \cdot \frac{\frac{\varepsilon}{\sin x \cdot \cos x}}{1 - \tan \varepsilon \cdot \tan x} \]
11. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\sin x \cdot \cos x} \cdot \frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (/ eps (* (sin x) (cos x))) (/ (tan x) (- 1.0 (* (tan eps) (tan x))))))

C

double code(double x, double eps) {
	return (eps / (sin(x) * cos(x))) * (tan(x) / (1.0 - (tan(eps) * tan(x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps / (sin(x) * cos(x))) * (tan(x) / (1.0d0 - (tan(eps) * tan(x))))
end function

Java

public static double code(double x, double eps) {
	return (eps / (Math.sin(x) * Math.cos(x))) * (Math.tan(x) / (1.0 - (Math.tan(eps) * Math.tan(x))));
}

Python

def code(x, eps):
	return (eps / (math.sin(x) * math.cos(x))) * (math.tan(x) / (1.0 - (math.tan(eps) * math.tan(x))))

Julia

function code(x, eps)
	return Float64(Float64(eps / Float64(sin(x) * cos(x))) * Float64(tan(x) / Float64(1.0 - Float64(tan(eps) * tan(x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps / (sin(x) * cos(x))) * (tan(x) / (1.0 - (tan(eps) * tan(x))));
end

Wolfram

code[x_, eps_] := N[(N[(eps / N[(N[Sin[x], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

   1. tan-sum N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1\right), \color{blue}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}\right)}\right) \]

4. Applied egg-rr 60.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{\mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{\sin x}{\cos x}\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{\mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin x, \cos x\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\mathsf{tan.f64}\left(x\right)}, \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \cos x\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(\color{blue}{x}\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \left(\frac{\cos x}{\sin x}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\cos x, \sin x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{\mathsf{tan.f64}\left(\varepsilon\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \sin x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\color{blue}{\varepsilon}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. sin-lowering-sin.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{cos.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{sin.f64}\left(x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(\varepsilon\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

7. Simplified 99.1%

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

   1. div-inv N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right)\right), \color{blue}{\left(\frac{1}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}}\right)}\right) \]

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \frac{\varepsilon}{\sin x \cdot \cos x} \cdot \frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} \]
11. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\varepsilon + x\right)\\ \mathbf{if}\;t\_0 - \tan x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\frac{-1}{\tan x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (tan (+ eps x))))
   (if (<= (- t_0 (tan x)) 2e-11)
     (+
      (* eps (+ 1.0 (* 0.3333333333333333 (* eps eps))))
      (*
       x
       (+
        (* eps (* x (+ 1.0 (* (* eps eps) 1.3333333333333333))))
        (* (* eps eps) (+ 1.0 (* (* eps eps) 0.6666666666666666))))))
     (+ t_0 (/ 1.0 (/ -1.0 (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan((eps + x));
	double tmp;
	if ((t_0 - tan(x)) <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = t_0 + (1.0 / (-1.0 / tan(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((eps + x))
    if ((t_0 - tan(x)) <= 2d-11) then
        tmp = (eps * (1.0d0 + (0.3333333333333333d0 * (eps * eps)))) + (x * ((eps * (x * (1.0d0 + ((eps * eps) * 1.3333333333333333d0)))) + ((eps * eps) * (1.0d0 + ((eps * eps) * 0.6666666666666666d0)))))
    else
        tmp = t_0 + (1.0d0 / ((-1.0d0) / tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan((eps + x));
	double tmp;
	if ((t_0 - Math.tan(x)) <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = t_0 + (1.0 / (-1.0 / Math.tan(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan((eps + x))
	tmp = 0
	if (t_0 - math.tan(x)) <= 2e-11:
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))))
	else:
		tmp = t_0 + (1.0 / (-1.0 / math.tan(x)))
	return tmp

Julia

function code(x, eps)
	t_0 = tan(Float64(eps + x))
	tmp = 0.0
	if (Float64(t_0 - tan(x)) <= 2e-11)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * Float64(eps * eps)))) + Float64(x * Float64(Float64(eps * Float64(x * Float64(1.0 + Float64(Float64(eps * eps) * 1.3333333333333333)))) + Float64(Float64(eps * eps) * Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / Float64(-1.0 / tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan((eps + x));
	tmp = 0.0;
	if ((t_0 - tan(x)) <= 2e-11)
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	else
		tmp = t_0 + (1.0 / (-1.0 / tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Tan[x], $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(x * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[(-1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 1.99999999999999988e-11

   1. Initial program 61.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)} \]

   if 1.99999999999999988e-11 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 62.4%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \left(\frac{\sin x}{\color{blue}{\cos x}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \left(\frac{1}{\color{blue}{\frac{\cos x}{\sin x}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\cos x}{\sin x}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\sin x}{\cos x}}}\right)\right)\right) \]

      5. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\tan x}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\tan x}\right)\right)\right) \]

      7. tan-lowering-tan.f64 62.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 62.4%

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\frac{1}{\tan x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) + \frac{1}{\frac{-1}{\tan x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\varepsilon + x\right) - \tan x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan (+ eps x)) (tan x))))
   (if (<= t_0 2e-11)
     (+
      (* eps (+ 1.0 (* 0.3333333333333333 (* eps eps))))
      (*
       x
       (+
        (* eps (* x (+ 1.0 (* (* eps eps) 1.3333333333333333))))
        (* (* eps eps) (+ 1.0 (* (* eps eps) 0.6666666666666666))))))
     t_0)))

C

double code(double x, double eps) {
	double t_0 = tan((eps + x)) - tan(x);
	double tmp;
	if (t_0 <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((eps + x)) - tan(x)
    if (t_0 <= 2d-11) then
        tmp = (eps * (1.0d0 + (0.3333333333333333d0 * (eps * eps)))) + (x * ((eps * (x * (1.0d0 + ((eps * eps) * 1.3333333333333333d0)))) + ((eps * eps) * (1.0d0 + ((eps * eps) * 0.6666666666666666d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan((eps + x)) - Math.tan(x);
	double tmp;
	if (t_0 <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan((eps + x)) - math.tan(x)
	tmp = 0
	if t_0 <= 2e-11:
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(Float64(eps + x)) - tan(x))
	tmp = 0.0
	if (t_0 <= 2e-11)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * Float64(eps * eps)))) + Float64(x * Float64(Float64(eps * Float64(x * Float64(1.0 + Float64(Float64(eps * eps) * 1.3333333333333333)))) + Float64(Float64(eps * eps) * Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan((eps + x)) - tan(x);
	tmp = 0.0;
	if (t_0 <= 2e-11)
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-11], N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(x * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 1.99999999999999988e-11

   1. Initial program 61.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)} \]

   if 1.99999999999999988e-11 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 62.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{{\sin x}^{2}}{\color{blue}{{\cos x}^{2}}}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({\sin x}^{2}\right), \color{blue}{\left({\cos x}^{2}\right)}\right)\right)\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\sin x, 2\right), \left({\color{blue}{\cos x}}^{2}\right)\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(x\right), 2\right), \left({\cos \color{blue}{x}}^{2}\right)\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\cos x, \color{blue}{2}\right)\right)\right)\right) \]

   10. cos-lowering-cos.f64 98.3%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), 2\right)\right)\right)\right) \]

5. Simplified 98.3%

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

Alternative 8: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) + \frac{1}{\cos x \cdot \frac{-1}{\sin x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2e-11)
   (+
    (* eps (+ 1.0 (* 0.3333333333333333 (* eps eps))))
    (*
     x
     (+
      (* eps (* x (+ 1.0 (* (* eps eps) 1.3333333333333333))))
      (* (* eps eps) (+ 1.0 (* (* eps eps) 0.6666666666666666))))))
   (+ (tan (+ eps x)) (/ 1.0 (* (cos x) (/ -1.0 (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = tan((eps + x)) + (1.0 / (cos(x) * (-1.0 / sin(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2d-11) then
        tmp = (eps * (1.0d0 + (0.3333333333333333d0 * (eps * eps)))) + (x * ((eps * (x * (1.0d0 + ((eps * eps) * 1.3333333333333333d0)))) + ((eps * eps) * (1.0d0 + ((eps * eps) * 0.6666666666666666d0)))))
    else
        tmp = tan((eps + x)) + (1.0d0 / (cos(x) * ((-1.0d0) / sin(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2e-11) {
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	} else {
		tmp = Math.tan((eps + x)) + (1.0 / (Math.cos(x) * (-1.0 / Math.sin(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2e-11:
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))))
	else:
		tmp = math.tan((eps + x)) + (1.0 / (math.cos(x) * (-1.0 / math.sin(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2e-11)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * Float64(eps * eps)))) + Float64(x * Float64(Float64(eps * Float64(x * Float64(1.0 + Float64(Float64(eps * eps) * 1.3333333333333333)))) + Float64(Float64(eps * eps) * Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666))))));
	else
		tmp = Float64(tan(Float64(eps + x)) + Float64(1.0 / Float64(cos(x) * Float64(-1.0 / sin(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2e-11)
		tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
	else
		tmp = tan((eps + x)) + (1.0 / (cos(x) * (-1.0 / sin(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2e-11], N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(x * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Cos[x], $MachinePrecision] * N[(-1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.99999999999999988e-11

   1. Initial program 61.0%

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

   3. Taylor expanded in eps around 0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)} \]

   if 1.99999999999999988e-11 < eps

   1. Initial program 62.4%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \left(\frac{\sin x}{\color{blue}{\cos x}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \left(\frac{1}{\color{blue}{\frac{\cos x}{\sin x}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\cos x}{\sin x}\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\sin x}{\cos x}}}\right)\right)\right) \]

      5. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\tan x}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\tan x}\right)\right)\right) \]

      7. tan-lowering-tan.f64 62.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 62.4%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\sin x}{\color{blue}{\cos x}}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\sin x} \cdot \color{blue}{\cos x}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{\sin x}\right), \color{blue}{\cos x}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \sin x\right), \cos \color{blue}{x}\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(x\right)\right), \cos x\right)\right)\right) \]

      6. cos-lowering-cos.f64 62.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(x\right)\right), \mathsf{cos.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 62.5%

      \[\leadsto \tan \left(x + \varepsilon\right) - \frac{1}{\color{blue}{\frac{1}{\sin x} \cdot \cos x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) + \frac{1}{\cos x \cdot \frac{-1}{\sin x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* eps (+ 1.0 (* 0.3333333333333333 (* eps eps))))
  (*
   x
   (+
    (* eps (* x (+ 1.0 (* (* eps eps) 1.3333333333333333))))
    (* (* eps eps) (+ 1.0 (* (* eps eps) 0.6666666666666666)))))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (1.0d0 + (0.3333333333333333d0 * (eps * eps)))) + (x * ((eps * (x * (1.0d0 + ((eps * eps) * 1.3333333333333333d0)))) + ((eps * eps) * (1.0d0 + ((eps * eps) * 0.6666666666666666d0)))))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
}

Python

def code(x, eps):
	return (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * Float64(eps * eps)))) + Float64(x * Float64(Float64(eps * Float64(x * Float64(1.0 + Float64(Float64(eps * eps) * 1.3333333333333333)))) + Float64(Float64(eps * eps) * Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + (0.3333333333333333 * (eps * eps)))) + (x * ((eps * (x * (1.0 + ((eps * eps) * 1.3333333333333333)))) + ((eps * eps) * (1.0 + ((eps * eps) * 0.6666666666666666)))));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(x * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

8. Simplified 97.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)} \]
9. Add Preprocessing

Alternative 10: 98.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(1 + x \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right) + \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* 0.3333333333333333 (* eps eps))
   (+
    1.0
    (*
     x
     (+
      (* x (+ 1.0 (* (* eps eps) 1.3333333333333333)))
      (* eps (+ 1.0 (* (* eps eps) 0.6666666666666666)))))))))

C

double code(double x, double eps) {
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + (x * ((x * (1.0 + ((eps * eps) * 1.3333333333333333))) + (eps * (1.0 + ((eps * eps) * 0.6666666666666666)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((0.3333333333333333d0 * (eps * eps)) + (1.0d0 + (x * ((x * (1.0d0 + ((eps * eps) * 1.3333333333333333d0))) + (eps * (1.0d0 + ((eps * eps) * 0.6666666666666666d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + (x * ((x * (1.0 + ((eps * eps) * 1.3333333333333333))) + (eps * (1.0 + ((eps * eps) * 0.6666666666666666)))))));
}

Python

def code(x, eps):
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + (x * ((x * (1.0 + ((eps * eps) * 1.3333333333333333))) + (eps * (1.0 + ((eps * eps) * 0.6666666666666666)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(0.3333333333333333 * Float64(eps * eps)) + Float64(1.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(Float64(eps * eps) * 1.3333333333333333))) + Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + (x * ((x * (1.0 + ((eps * eps) * 1.3333333333333333))) + (eps * (1.0 + ((eps * eps) * 0.6666666666666666)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x * N[(N[(x * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + 1\right) + \color{blue}{x} \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \color{blue}{\left(1 + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(1 + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{1} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right), \left(1 + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(1 + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(1 + x \cdot \left(x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right) + \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 11: 97.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(1 + \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right) \cdot \left(\varepsilon \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* 0.3333333333333333 (* eps eps))
   (+ 1.0 (* (+ 1.0 (* (* eps eps) 0.6666666666666666)) (* eps x))))))

C

double code(double x, double eps) {
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + ((1.0 + ((eps * eps) * 0.6666666666666666)) * (eps * x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((0.3333333333333333d0 * (eps * eps)) + (1.0d0 + ((1.0d0 + ((eps * eps) * 0.6666666666666666d0)) * (eps * x))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + ((1.0 + ((eps * eps) * 0.6666666666666666)) * (eps * x))));
}

Python

def code(x, eps):
	return eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + ((1.0 + ((eps * eps) * 0.6666666666666666)) * (eps * x))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(0.3333333333333333 * Float64(eps * eps)) + Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(eps * eps) * 0.6666666666666666)) * Float64(eps * x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((0.3333333333333333 * (eps * eps)) + (1.0 + ((1.0 + ((eps * eps) * 0.6666666666666666)) * (eps * x))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{\varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + 1\right) + \color{blue}{\varepsilon} \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \color{blue}{\left(1 + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(1 + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{1} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right), \left(1 + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(1 + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{\left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot x\right), \color{blue}{\left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \left(\color{blue}{1} + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{2}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{2}{3}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 96.7%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{2}{3}\right)\right)\right)\right)\right)\right) \]

8. Simplified 96.7%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(1 + \left(\varepsilon \cdot x\right) \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right)\right)\right)} \]

9. Final simplification 96.7%

   \[\leadsto \varepsilon \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(1 + \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666\right) \cdot \left(\varepsilon \cdot x\right)\right)\right) \]
10. Add Preprocessing

Alternative 12: 97.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.3333333333333333 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.13333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    (* eps eps)
    (+ 0.3333333333333333 (* (* eps eps) 0.13333333333333333))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * (0.3333333333333333 + ((eps * eps) * 0.13333333333333333))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((eps * eps) * (0.3333333333333333d0 + ((eps * eps) * 0.13333333333333333d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * (0.3333333333333333 + ((eps * eps) * 0.13333333333333333))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((eps * eps) * (0.3333333333333333 + ((eps * eps) * 0.13333333333333333))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * Float64(0.3333333333333333 + Float64(Float64(eps * eps) * 0.13333333333333333)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((eps * eps) * (0.3333333333333333 + ((eps * eps) * 0.13333333333333333))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(eps * eps), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\sin \varepsilon, \color{blue}{\cos \varepsilon}\right) \]

   2. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\varepsilon\right), \cos \color{blue}{\varepsilon}\right) \]

   3. cos-lowering-cos.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\varepsilon\right), \mathsf{cos.f64}\left(\varepsilon\right)\right) \]

5. Simplified 96.7%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{3}} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{3}} + \frac{2}{15} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{2}{15} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{2}{15}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 96.7%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{2}{15}\right)\right)\right)\right)\right) \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.3333333333333333 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.13333333333333333\right)\right)} \]
9. Add Preprocessing

Alternative 13: 97.8% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* 0.3333333333333333 (* eps eps)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (0.3333333333333333 * (eps * eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (0.3333333333333333d0 * (eps * eps)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (0.3333333333333333 * (eps * eps)));
}

Python

def code(x, eps):
	return eps * (1.0 + (0.3333333333333333 * (eps * eps)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * Float64(eps * eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (0.3333333333333333 * (eps * eps)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\left(\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -0.16666666666666666\right) - \left(-0.5 + \left(\frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \varepsilon \cdot \left(\frac{\sin x \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 96.7%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
9. Add Preprocessing

Alternative 14: 97.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\sin \varepsilon, \color{blue}{\cos \varepsilon}\right) \]

   2. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\varepsilon\right), \cos \color{blue}{\varepsilon}\right) \]

   3. cos-lowering-cos.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\varepsilon\right), \mathsf{cos.f64}\left(\varepsilon\right)\right) \]

5. Simplified 96.7%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon} \]
7. Step-by-step derivation

8. Simplified 96.7%

   \[\leadsto \color{blue}{\varepsilon} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024155 
(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 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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